If a PCF converges to an integer or a rational, I won't post about it here. Too boring. Also, all of these relationships are conjectural; I don't have any proofs for you. There could be false positives. But when you see 1.7182818284590453, you think "{e - 1}" and not "probably just a number that looks like {e - 1} to 16 decimal places. Also, I haven't found symbolic expressions for some of the PCF values below, and maybe that's because the PCFs don't converge, and then the values that I get for the PCF after calculating 1000 terms won't be anything meaningful, just a pitstop on the way to nowhere. But a bunch of the ones I don't have symbolic expressions for are clearly related to each other (by Mobius transformations), so those ones at least are probably converging to something with a simple algorithmic structure that can be expressed using different mathematics. At some point, I'll check for non-convergence, and remove any unruly PCFs from the list below.
Edit: I think I've removed all the non-convergent values, but there will still be some false-positive results in places where I didn't check equality to high enough precision.
If you have continued fractions with constants in the numerator and denominators, you just get the boring values (ℤ, ℚ, and the quadratics) when the fraction converges to any value at all (i.e. when the denominators are non-zero). Here below are some of the (0th order over 0th order) PCFs that converge to quadratic irrational numbers. When I try to find an expression for the value of a PCF, I look to see if the value is a Möbius transformation of a famous mathematical constant.
So for example when I say {K:1/1 = 1.618033988749895 : ['sqrt(5):', [1, 1, 0, 2]]}, I mean that the continued fraction with 1s for numerators and 1s for denominators converges to a value of (1 * sqrt(5) + 1) / (0 * sqrt(5) + 2).
K:-1/3 = 2.618033988749895 : ['sqrt(5)', [1, 1, 1, -1]]
K:-1/4 = 3.732050807568877 : ['sqrt(3)', [1, 2, 0, 1]]
K:-1/5 = 4.79128784747792 : ['sqrt(21)', [1, 5, 0, 2]]
K:-1/6 = 5.82842712474619 : ['sqrt(2)', [1, 1, 1, -1]]
K:-1/7 = 6.854101966249685 : ['sqrt(5)', [2, 4, 1, -1]]
K:-1/8 = 7.872983346207417 : ['sqrt(15)', [1, 4, 0, 1]]
K:-1/9 = 8.88748219369606 : ['sqrt(77)', [1, 7, 1, -7]]
K:-1/10 = 9.898979485566356 : ['sqrt(6)', [1, 2, 1, -2]]
K:-2/4 = 3.414213562373095 : ['sqrt(2)', [1, 0, 1, -1]]
K:-2/5 = 4.56155281280883 : ['sqrt(17)', [1, 1, 1, -3]]
K:-2/6 = 5.645751311064591 : ['sqrt(7)', [1, 3, 0, 1]]
K:-2/7 = 6.701562118716424 : ['sqrt(41)', [1, 3, 1, -5]]
K:-2/8 = 7.741657386773942 : ['sqrt(14)', [1, 4, 0, 1]]
K:-2/9 = 8.772001872658766 : ['sqrt(73)', [1, 5, 1, -7]]
K:-2/10 = 9.79583152331272 : ['sqrt(23)', [1, 5, 0, 1]]
K:-3/5 = 4.302775637731995 : ['sqrt(13)', [1, -1, 1, -3]]
K:-3/6 = 5.449489742783178 : ['sqrt(6)', [1, 0, 1, -2]]
K:-3/7 = 6.54138126514911 : ['sqrt(37)', [1, 1, 1, -5]]
K:-3/8 = 7.60555127546399 : ['sqrt(13)', [1, 4, 0, 1]]
K:-3/9 = 8.653311931459037 : ['sqrt(69)', [1, 3, 1, -7]]
K:-3/10 = 9.690415759823429 : ['sqrt(22)', [1, 5, 0, 1]]
K:-4/6 = 5.23606797749979 : ['sqrt(5)', [1, 3, 0, 1]]
K:-4/7 = 6.372281323269014 : ['sqrt(33)', [1, -1, 1, -5]]
K:-4/8 = 7.464101615137754 : ['sqrt(3)', [0, 2, -1, 2]]
K:-4/9 = 8.531128874149275 : ['sqrt(65)', [1, 1, 1, -7]]
K:-4/10 = 9.58257569495584 : ['sqrt(21)', [1, 5, 0, 1]]
K:-5/5 = 3.618033988749895 : ['sqrt(5)', [2, 0, 1, -1]]
K:-5/7 = 6.192582403567252 : ['sqrt(29)', [2, 4, 1, -3]]
K:-5/8 = 7.3166247903554 : ['sqrt(11)', [1, 4, 0, 1]]
K:-5/9 = 8.405124837953327 : ['sqrt(61)', [1, -1, 1, -7]]
K:-5/10 = 9.47213595499958 : ['sqrt(5)', [1, 0, 1, -2]]
K:-6/6 = 4.732050807568878 : ['sqrt(3)', [2, 0, 1, -1]]
K:-6/8 = 7.162277660168379 : ['sqrt(10)', [1, 4, 0, 1]]
K:-6/9 = 8.274917217635375 : ['sqrt(57)', [1, -3, 1, -7]]
K:-6/10 = 9.358898943540673 : ['sqrt(19)', [1, 5, 0, 1]]
K:-7/6 = 4.414213562373095 : ['sqrt(2)', [2, -1, 1, -1]]
K:-7/7 = 5.79128784747792 : ['sqrt(21)', [2, 0, 1, -3]]
K:-7/9 = 8.140054944640259 : ['sqrt(53)', [2, 4, 1, -5]]
K:-7/10 = 9.242640687119286 : ['sqrt(2)', [2, 1, 1, -1]]
K:-8/7 = 5.56155281280883 : ['sqrt(17)', [2, -2, 1, -3]]
K:-8/8 = 6.82842712474619 : ['sqrt(2)', [2, 0, 1, -1]]
K:-8/10 = 9.12310562561766 : ['sqrt(17)', [1, 5, 0, 1]]
K:-9/7 = 5.302775637731995 : ['sqrt(13)', [3, 3, 1, -1]]
K:-9/8 = 6.645751311064591 : ['sqrt(7)', [2, -1, 1, -2]]
K:-9/9 = 7.854101966249685 : ['sqrt(5)', [3, 3, 1, -1]]
K:-10/8 = 6.449489742783178 : ['sqrt(6)', [1, 4, 0, 1]]
K:-10/9 = 7.701562118716424 : ['sqrt(41)', [2, -2, 1, -5]]
K:-10/10 = 8.872983346207416 : ['sqrt(15)', [2, 0, 1, -3]]
K:1/1 = 1.618033988749895 : ['sqrt(5)', [1, 1, 0, 2]]
K:1/2 = 2.414213562373095 : ['sqrt(2)', [1, 1, 0, 1]]
K:1/3 = 3.302775637731995 : ['sqrt(13)', [1, 3, 0, 2]]
K:1/4 = 4.23606797749979 : ['sqrt(5)', [1, 2, 0, 1]]
K:1/5 = 5.192582403567252 : ['sqrt(29)', [1, 5, 0, 2]]
K:1/6 = 6.162277660168379 : ['sqrt(10)', [1, 3, 0, 1]]
K:1/7 = 7.1400549446402595 : ['sqrt(53)', [0, 2, 1, -7]]
K:1/8 = 8.12310562561766 : ['sqrt(17)', [1, 4, 0, 1]]
K:1/9 = 9.109772228646444 : ['sqrt(85)', [1, 9, 0, 2]]
K:1/10 = 10.099019513592784 : ['sqrt(26)', [1, 5, 0, 1]]
K:2/2 = 2.732050807568877 : ['sqrt(3)', [1, 1, 0, 1]]
K:2/3 = 3.5615528128088303 : ['sqrt(17)', [1, 3, 0, 2]]
K:2/4 = 4.449489742783178 : ['sqrt(6)', [1, 2, 0, 1]]
K:2/5 = 5.372281323269014 : ['sqrt(33)', [1, 5, 0, 2]]
K:2/6 = 6.3166247903554 : ['sqrt(11)', [1, 3, 0, 1]]
K:2/7 = 7.274917217635375 : ['sqrt(57)', [-1, -7, 0, -2]]
K:2/8 = 8.242640687119286 : ['sqrt(2)', [1, 2, 1, -1]]
K:2/9 = 9.216990566028302 : ['sqrt(89)', [1, 9, 0, 2]]
K:2/10 = 10.196152422706632 : ['sqrt(3)', [1, 1, -1, 2]]
K:3/1 = 2.302775637731995 : ['sqrt(13)', [1, 1, 0, 2]]
K:3/3 = 3.79128784747792 : ['sqrt(21)', [1, 3, 0, 2]]
K:3/4 = 4.645751311064591 : ['sqrt(7)', [1, 2, 0, 1]]
K:3/5 = 5.54138126514911 : ['sqrt(37)', [1, 5, 0, 2]]
K:3/6 = 6.464101615137754 : ['sqrt(3)', [1, 0, -1, 2]]
K:3/7 = 7.405124837953327 : ['sqrt(61)', [1, 7, 0, 2]]
K:3/8 = 8.358898943540673 : ['sqrt(19)', [1, 4, 0, 1]]
K:3/9 = 9.321825380496477 : ['sqrt(93)', [1, 9, 0, 2]]
K:3/10 = 10.291502622129181 : ['sqrt(7)', [1, 1, -1, 3]]
K:4/1 = 2.5615528128088303 : ['sqrt(17)', [1, 1, 0, 2]]
K:4/2 = 3.23606797749979 : ['sqrt(5)', [1, 1, 0, 1]]
K:4/4 = 4.82842712474619 : ['sqrt(2)', [0, 2, 1, -1]]
K:4/5 = 5.701562118716424 : ['sqrt(41)', [1, 5, 0, 2]]
K:4/6 = 6.60555127546399 : ['sqrt(13)', [1, 3, 0, 1]]
K:4/7 = 7.531128874149275 : ['sqrt(65)', [1, 7, 0, 2]]
K:4/8 = 8.47213595499958 : ['sqrt(5)', [0, 2, 1, -2]]
K:4/9 = 9.424428900898052 : ['sqrt(97)', [1, 9, 0, 2]]
K:4/10 = 10.385164807134505 : ['sqrt(29)', [1, 5, 0, 1]]
The constants that have come up as useful so far for PCFs of degree (1 over 1) are:
- e
- e^2
- sqrt(e)
- tan(1)
- sqrt(2 / e pi)) / erfc(1/sqrt(2))
- sqrt(pi e / 2) * erf(1/sqrt(2))
and a bunch of ratios of Bessel functions of the first kind, jv(,)/jv(,) and ratios of modified Bessel function of the first kind iv(,)/iv(,). The names "jv" and "iv" are not standard in mathematics, but they're the function names that scipy.special uses and I like them fine. I've tested linear transforms of elementary functions of hundred of constants and I'm hoping that others will show up later, but those constants above are the only ones that have made an appearance so far.
Here are 140 simple PCFs (1st order over 1st order, small coefficients), and estimates of their values using 1000 fractional terms, and alternative symbolic expressions for the values when I have them. Again, the symbolic expressions give a constant and four numbers, which are the coefficients of a Mobius transformation from the constant to the PCF value. When I say that K:1/(1 + -2n) equals (e^2, [0, 2, 1, 1]), that's just means (0*e^2 + 2)/(1 * e^2 + 1), better known as 2/(e^2+1). This doesn't look very compressed for the case of e^2, but when the constant is e.g. sqrt(pi * e / 2) * erf(sqrt(1/2)), then the (constant, [transformation coefficients]) presentation-form starts to make more sense.
K:(-1 + -2n)/(2 + n) = 5.163306117610533 : ['iv(2, 1)/iv(1, 1)', [1, 1, 1, 0]]
K:(-1 + -2n)/-n = 0.7336562064501501 : ['iv(2, 1)/iv(1, 1)', [5, 1, 0, 3]]
K:(-1 + 2n)/(1 + -n) = 0.1936743584869569 : ['iv(2, 1)/iv(1, 1)', [1, 0, 1, 1]]
K:(-1 + 2n)/(1 + n) = 1.36303624396306 : ['iv(1, 1)/iv(2, 1)', [3, 0, 1, 5]]
K:(-1 + 2n)/(2 + -n) = 0.5536100341034655 : ['iv(0, 1)/iv(1, 1)', [1, -1, 1, 0]]
K:(1 + -2n)/(-1 + n) = 0.36303624396306 : ['iv(2, 1)/iv(1, 1)', [5, -2, -5, -1]]
K:(1 + -2n)/(1 + n) = 0.8063256415130431 : ['iv(1, 1)/iv(2, 1)', [1, 0, 1, 1]]
K:(1 + -2n)/(2 + 2n) = 1.7088749052272068 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [1, 1, 1, 0]]
K:(1 + -2n)/(2 + n) = 1.4463899658965345 : ['iv(0, 1)/iv(1, 1)', [1, 1, 1, 0]]
K:(1 + -2n)/n = 0.5132482562519421 : ['iv(1, 1)/iv(2, 1)', [1, -1, 1, 2]]
K:(1 + 2n)/(-1 + n) = 0.6913764776985003 : ['iv(0, 1)/iv(1, 1)', [1, 0, 1, 1]]
K:(1 + 2n)/(-2 + n) = 0.37155708548628574 : ['iv(0, 1)/iv(1, 1)', [1, 1, 3, 2]]
K:(1 + 2n)/(1 + n) = 1.9483748611298197 : ['iv(2, 1)/iv(1, 1)', [2, 1, -1, 1]]
K:(1 + 2n)/(2 + n) = 2.7545459072724148 : ['iv(2, 1)/iv(1, 1)', [5, 1, -5, 2]]
K:(1 + 2n)/n = 1.2401937238700897 : ['iv(1, 1)/iv(2, 1)', [1, 1, 1, 0]]
K:(1 + n)/(1 + n) = 1.7182818284590453 : ['e', [1, -1, 0, 1]]
K:(1 + n)/(2 + n) = 2.549646778303845 : ['e', [1, -2, -1, 3]]
K:(1 + n)/1 = 1.9042712333296918 : ['sqrt(2/(e pi)) / erfc(1/sqrt(2))', [0, 1, 1, -1]]
K:(2 + 2n)/(-2 + 2n) = 0.3264440198011198 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [2, 0, 4, 3]]
K:(2 + 2n)/(1 + n) = 2.194528049465325 : ['e^2', [1, -3, 0, 2]]
K:(2 + 2n)/(2 + 2n) = 2.821372269284896 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [2, 0, 0, 1]]
K:(2 + 2n)/2n = 1.2616860984829235 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [2, 2, 2, 1]]
K:(2 + 2n)/n = 1.5231883119115297 : ['e^2', [2, -2, 1, 1]]
K:(2 + n)/(2 + n) = 2.7844223823546654 : ['e', [0, 2, 1, -2]]
K:(2 + n)/1 = 2.2117257812521967 : ['sqrt(2/(e pi)) / erfc(1/sqrt(2))', [2, -2, -1, 2]]
K:-1/(-2n) = 0.575080915004306 : ['jv(1, 1)/jv(0, 1)', [1, 0, 0, 1]]
K:-1/(1 + -2n) = 2.557407724654902 : ['tan(1)', [1, 1, 0, 1]]
K:-1/(1 + -n) = 0.6117892344322042 : ['jv(2, 2)/jv(1, 2)', [1, 0, 0, 1]]
K:-1/(1 + 2n) = 0.6420926159343308 : ['tan(1)', [0, 1, 1, 0]]
K:-1/(1 + n) = 0.38821076556779577 : ['jv(0, 2)/jv(1, 2)', [1, 0, 0, 1]]
K:-1/(2 + -2n) = 0.2611142642552964 : ['jv(2, 1)/jv(1, 1)', [1, 0, 0, 1]]
K:-1/(2 + -n) = 0.365450152243868 : ['jv(2, 2)/jv(0, 2)', [1, -1, 1, 0]]
K:-1/(2 + 2n) = 1.7388857357447036 : ['jv(0, 1)/jv(1, 1)', [1, 0, 0, 1]]
K:-1/(2 + n) = 1.634549847756132 : ['jv(1, 2)/jv(2, 2)', [1, 0, 0, 1]]
K:-1/-n = 2.5759203213682222 : ['jv(1, 2)/jv(0, 2)', [1, 0, 0, 1]]
K:-n/(2 + n) = 1.581976706869326 : ['e', [1, 0, 1, -1]]
K:1/(-1 + n) = 0.4331274267223117 : ['iv(2, 2)/iv(1, 2)', [1, 0, 0, 1]]
K:1/(-2 + 2n) = 0.24019372387008975 : ['iv(2, 1)/iv(1, 1)', [1, 0, 0, 1]]
K:1/(-2 + n) = 0.30878937306624 : ['iv(2, 2)/iv(1, 2)', [2, -1, -1, 0]]
K:1/(1 + -2n) = 0.23840584404423504 : ['e^2', [0, 2, 1, 1]]
K:1/(1 + 2n) = 1.313035285499331 : ['e^2', [1, 1, 1, -1]]
K:1/(1 + n) = 1.4331274267223117 : ['iv(0, 2)/iv(1, 2), ', [1, 0, 0, 1]]
K:1/(2 + 2n) = 2.2401937238700897 : ['iv(0, 1)/iv(1, 1)', [1, 0, 0, 1]]
K:1/(2 + n) = 2.30878937306624 : ['iv(1, 2)/iv(2, 2)', [1, 0, 0, 1]]
K:1/2n = 0.4463899658965345 : ['iv(1, 1)/iv(0, 1)', [1, 0, 0, 1]]
K:1/n = 0.697774657964008 : ['iv(1, 2)/iv(0, 2)', [1, 0, 0, 1]]
K:2n/(-1 + n) = 0.313035285499331 : ['e^2', [0, 2, 1, -1]]
K:2n/(1 + 2n) = 1.5414940825367982 : ['sqrt(e)', [0, 1, 1, -1]]
K:2n/(1 + n) = 1.6743014120892405 : ['e^2', [0, 4, 1, -5]]
K:2n/(2 + -2n) = 0.4148196588637698 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [0, 1, 1, 1]]
K:2n/(2 + 2n) = 2.434949504371802 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [0, 1, 1, -1]]
K:2n/2n = 0.7088749052272068 : ['sqrt(pi * e / 2) * erf(1/sqrt(2))', [0, 1, 1, 0]]
K:2n/n = 0.9113576837112107 : ['e^2', [0, 4, 1, -3]]
K:n/(1 + n) = 1.3922111911773327 : ['e', [0, 1, 1, -2]]
K:n/(2 + n) = 2.2906166927853624 : ['e', [0, 1, 2, -5]]
K:n/1 = 1.5251352761609813 : ['sqrt(2/(e pi)) / erfc(1/sqrt(2))', [1, 0, 0, 1]]
K:n/n = 0.581976706869326 : ['e', [0, 1, 1, -1]]
I haven't found alternative representations for about 3/5s of them yet! I will call those the... mysterious PCFs. Some of the mysterious PCF values are related to at least one other mysterious PCF value by a Mobius transformation, so if I can figure out an alternative representation for one, then I think I should be able to find an alternative representation for a few more right away. They come in families.
I haven't really looked at how the polynomials across PCFs are related when their values are related by a Mobius transformation. That could be fun. A lot of the values have the same decimal part and the Mobius transformation is just changing the integer.
Also, going to higher degrees might be fun. I'll probably do that next. I haven't made any real progress on the remaining 2/3s for a day and I could use a pick-me-up, like a new formula for zeta(5) or acosh(2) or 1/(sqt(2)*csc(sqrt(2)) - 1) or something.
Oof, my programs are finding expressions for far fewer than 1/3 of the quadratic PCFs. There was a pi expression that showed up, which was nice, but this shall take some time and ingenuity.
Here's my ingenious plan: include integrals. There's a famous constant called the Euler-Gompertz constant which has simple representations in terms of polynomial continued fractions and in terms of multiple simple integrals, and likewise for other constants like pi and e. Maybe I can find some new integrals that describe the mysterious PCF values! I guess I'll... generate a few thousand simple functions, evaluate their integrals from [0, 1] or [-1, 1], or [0, inf] or something and then compare against the PCFs. Tomorrow or the next day or in two days. Soon. Soon! *crack of thunder* Ahahahahah.
Okay, so, same day, I did that, and it failed! Kind of. I generated 7,481 expressions-in-one-variable to use as integrands, and I integrated them from [0, 1] with SciPy, and 5,777 of them made it through the integration without errors like !divide-by-zero or !overflow or !expected-a-real-number-not-a-complex-one-you-dummy, and 5,780 is a great big number and I'm very happy with it. And then I looked at the decimal parts of the values of the 89 mystery PCFs above, and looked for exact matches among the integral values (or rather, exact matches up to the first 10 digits), and there were none! None! Very sad. Maybe I just need to look for Mobius transformations and then it will start working, but I thought I'd get some exact values without the transformation, and I didn't. Some of the generated integral values are things like {e} and they would have matched the PCF values that weren't mysterious, but I'm not after new integrals for e, I'm after integrals that alternatively characterize the remaining mysterious PCFs that are (1st order over 1st order) with small coefficients. Those are my white whales and I will keep after them. But not tonight, probably. I sure hope the linear-transformation things works. If I've got 5,777 numbers, and that increases by like, I don't know, a factor of 600 when I do the Mobius transformations, that's a lot of numbers. And a lot of numbers means a lot of time spent and a lot more chances for false positives, which then take time to rule out. I sure hope things start working out tomorrow and don't just take lots of time for no reason.
The Euler-Gompertz constant was one of the 5,777 that made it through, so at least I know that some of the integral values will be useful with the quadratic PCFs.
Not all of the integrals had distinct values. After getting rid of the repeats and the rationals, the set of 5,777 is more like 4,300 interesting irrationals. When I'm all done with this, I should totally take the integral code and the PCF code and extend the precision with mpmath and publish a little collection of real numbers. There are already a few collections online like that, but I think I'll have something of value to add. And it'll be in plain text instead of a pdf or an html table or something, because I'm a man of good taste and practicality.
Oh, neat, way fewer values made it through when I integrated the same equations from [-1, 1]. Only 2,664. That's not 2,664 unique values; only 2,664 equations could be evaluated at all. Again, no direct matches to the remaining first order PCF values without doing the Mobius transformations.
I tried Mobius transformations on just a few thousand of the [0, 1] integral values for a quick test, and no luck so far. *scrunches face in determination* But I will solve this, one way or another.
Okay, Mobius transformations of the full set of integral values doenn't seem to be working either. A tiny fraction of them of are related to (2nd/2nd order) PCF values, but I'm not making progress on the ones that are important to me. But I have a new idea; the incomplete gamma function! ... No that didn't help either. Maybe I'll just give up. Or no! No, I'll keep trying. I have spirit in me yet. ...
I think I'm going to focus my efforts on these six:
T1.) K:n/2n = 0.40848432946968583081519573053105789080
T2.) K:2/2n = 0.81204094122269147634277999564910728345
T3.) K:2/n = 1.126357239623422770851505807407127106552
T4.) K:2n/1 = 1.83270564129869841627395897237131435892
T5.) K:n/2 = 2.373215532822840867299032690826535012410
T6.) K:2n/2 = 2.63896751423479126047150115207156112883
If I can figure out alternative symbolic expressions for those by the end of the month, I'll be quite happy.
Ooh, I could look for patterns in the canonical continued fractions for those numbers next. I doubt that will help me find alternative symbolic expressions, but it sounds fun and interesting. Like tan(1) has shown up, and it's canonical CF has alternating 1s and odd numbers: [1; 1, 1, 3, 1, 5, 1, 7, 1, 9, 1, 11, 1, 13, 1, 15, 1, 17, 1, 19, 1, 21, 1, 23, 1, 25, 1, 2, ...]. And e has a nice one too: two ones separated by even numbers: [2;1,2,1,1,4,1,1,6,1,1,8,...].
Wow. No patterns in the canonical CFs for those six constants. Just small values separated by very very large values. Things like {... 2, 1, 1, 3, 1, 1, 1, 26, 2, 1, 6135446304522958095533595604256982673754985678302201104701055945259472746674739234940653896378678420905557664533716655240113567631620870212472857404594535214186749957401785, 1, 4, 2, ...}. Just a random 172 digit number in between the small ones. All 6 are like that.
I've spent a little time trying to find expressions for constants using RIES (Robert Munafo's inverse equation solver) with no luck, but maybe some of the six values above will work out. ...
Another possible resource is http://www.sequencedb.net, which hasn't helped me in the past, but seems really good, and maybe it will help with one or more of the six values above. ... Nope. No luck.
Maybe some Mobius transform of one of those 6 constants is more google-able. For example, {1/T1} is the convergent value of K:(1 + n)/(2 + 2n) = 2.448074327106375. The fraction is more complicated, but maybe the {1/T1} shows up in other parts of mathematics when {T1} doesn't. Spoilers: it doesn't. But I'll keep looking.
K:(1 + n)/(2 + 2n) = 2.448074327106375 : ['T1', [0, 1, 1, 0]]
K:(1 + n)/2n = 0.754403355133908 : ['T1', [2, 1, 1, 2]]
K:(2 + n)/(2 + 2n) = 2.6511016770928815 : ['T1', [-2, -4, -2, -1]]
K:n/(2 + -2n) = 0.6744491614535594 : ['T1', [3, 0, 2, 1]]
K:n/(2 + 2n) = 2.231772586610602 : ['T1', [1, 0, -2, 1]]
K:2/(-2 + 2n) = 0.4629300057070971 : ['T2', [2, -2, -1, 0]]
K:2/(2 + 2n) = 2.462930005707097 : ['T2', [0, 2, 1, 0]]
K:2/(-1 + n) = 0.7756355884645123 : ['T3', [1, -2, -1, 0]]
K:2/(1 + n) = 1.7756355884645123 : ['T3', [0, 2, 1, 0]]
K:2/(2 + n) = 2.5785304719698354 : ['T3', [2, 0, -1, 2]]
K:2/(3 + n) = 3.457034844146083 : ['T3', [1, -2, -2, 2]]
K:(1 + n)/2 = 2.679416883955586 : ['T5', [0, 1, 1, -2]]
K:(2 + 2n)/2 = 3.130049580681956 : ['T6', [0, 2, 1, -2]]
Haven't found any of those yet. Actually, google says one of them appears twice in an old Indian math text book from the late 1800s, but I didn't myself see the number when skimming the tables full of numbers and I didn't understand what the page was about, and the other numbers on the page weren't ... it just didn't seem right, okay?
Here are few more (1st/1st) PCFs with larger coefficients and with values that aren't ratios of Bessel functions:
K:(2 + 2n)/(3 + 2n) = 3.69348449872319 : ['sqrt(e)', [2, -2, -1, 2]]
K:(3 + -2n)/2n = 0.5851803411362303 : ['sqrt(pi * e / 2) * erf(sqrt(1/2))', [1, 0, 1, 1]]
K:-1/(3 + -2n) = 2.608979049230431 : ['tan(1)', [3, 2, 1, 1]]
K:-1/(3 + 2n) = 2.79401891249195 : ['tan(1)', [1, 0, 1, -1]]
K:1/(3 + 2n) = 3.194528049465325 : ['e^2', [1, -1, 0, 2]]
K:2/(-3 + n) = 0.4570348441460834 : ['sqrt(2 pi)', [2, 3, 5, 5]]
K:2n/(3 + 2n) = 3.361993867092269 : ['sqrt(e)', [0, 1, 2, -3]]
K:3n/(-2 + n) = 0.15718708947376786 : ['e^3', [0, 3, 1, -1]]
K:3n/(2 + 3n) = 2.527726473157129 : ['e^(1/3)', [0, 1, 1, -1]]
So now we've also got expressions relating to the constants sqrt(2 pi), e^3, and e^(1/3). A little exciting!
So... part of the reason I switched from looking for a PCF for 2^(1/3) to looking for expressions for PCF values was that I thought there might not be one for 2^(1/3) , and if I could say something about what sorts of values PCFs do converge to, then I could be like, "Not only is 2^(1/3) not a PCF value, but non-trivial roots higher than two of any integer will not be PCF values", and that would be cool. But I'm doing such a poor job of exhaustively characterizing the values of small PCFs, I think I might need another tactic. Maybe next I will find PCFs for any constants I can, and I'll make a list of those. First, second, third, fourth, any order, if I can find an expression for a constant, then the constant goes in a list. And then maybe the set of things in the list will give me some ideas for the mysterious PCFs which I still do want to know more about.
Hold up! I went down with the polynomial degree (1st/0th and 0th/1st instead of 1st/1st) and up with the coefficient size (-4 to 4), and I've got a bunch of new results! Or, you know, new to me anyway. For all I know, everything in this post was known to Euler and Gauss.
K:-1/(2 + -4n) = 2.5463024898437903 : ['tan(1/2)', [1, 2, 0, 1]]
K:-1/(2 + 4n) = 1.830487721712452 : ['tan(1/2)', [0, 1, 1, 0]]
K:-4/(-1 + -2n) = 0.9153151087205715 : ['tan(2)', [0, 2, -1, 0]]
K:-4/(-1 + 2n) = 3.3700797265230378 : ['tan(2)', [2, 1, 0, -1]]
K:-4/(2 + -4n) = 5.114815449309804 : ['tan(1)', [2, 2, 0, 1]]
K:-4/(2 + 4n) = 1.2841852318686615 : ['tan(1)', [0, 2, 1, 0]]
K:-4/(3 + 2n) = 2.088429199867794 : ['tan(2)', [4, 0, 1, -2]]
K:1/(2 + -4n) = 1.5378828427399902 : ['e', [1, 3, 1, 1]]
K:1/(2 + 4n) = 2.163953413738653 : ['e', [1, 1, 1, -1]]
K:2/(-4 + n) = 0.3760339624362077 : ['sqrt(2 pi)', [2, -2, 2, 3]]
K:4/(-1 + 2n) = 0.9280551601516336 : ['e^4', [1, -3, 1, 1]]
K:4/(1 + 2n) = 2.0746294414550963 : ['e^4', [2, 2, 1, -1]]
K:4/(2 + -4n) = 0.4768116880884701 : ['e^2', [0, 4, 1, 1]]
K:4/(2 + 4n) = 2.6260705709986625 : ['e^2', [2, 2, 1, -1]]
K:4/(3 + 2n) = 3.722213300413417 : ['e^4', [4, -4, 1, 3]]
K:4n/2 = 3.0502705523219626 : ['sqrt(2/(e pi)) / erfc(1/sqrt(2))', [2, 0, 0, 1]]
K:(4 + 4n)/2 = 3.8085424666593837 : ['sqrt(2/(e pi)) / erfc(1/sqrt(2))', [0, 2, 1, -1]]
Now we've got tan(2), tan(1/2), another sqrt(2 pi), an a bunch of e^4s! I was honestly concerned that we had tan(1) before but no others tan() values, but now, things are making more sense. Other rational values show up too. Wanna try (-5, 5) for coefficients next? ' Course you do!
Surprise, I went up to 6!
K:1/(3 + 6n) = 3.110296679619444 : ['e^(2/3)', [1, 1, 1, -1]]
K:1/(6 + 4n) = 6.09929355660769 : ['e', [1, -1, -1, 3]]
K:4/(6 + 4n) = 6.38905609893065 : ['e^2', [1, -1, 0, 1]]
K:-1/(5 + 2n) = 4.854814643897434 : ['tan(1)', [-1, 1, -2, 3]]
K:-1/(6 + 4n) = 5.899277681252529 : ['tan(1/2)', [1, 0, 2, -1]]
K:-1/(3 + -6n) = 3.3462535495105756 : ['tan(1/3)', [1, 3, 0, 1]]
K:1/(3 + -6n) = 2.6784872624683658 : ['e^(2/3)', [2, 4, 1, 1]]
K:-4/(6 + 4n) = 5.5880378249839 : ['tan(1)', [2, 0, 1, -1]]
K:-4/(3 + -6n) = 4.573685778945954 : ['tan(2/3)', [2, 3, 0, 1]]
K:-1/(3 + 6n) = 2.888057036277277 : ['tan(1/3)', [0, 1, 1, 0]]
K:-4/(3 + 6n) = 2.5418034867667014 : ['tan(2/3)', [0, 2, 1, 0]]
K:2/(-5 + n) = 0.31866852409452306 : ['tan(1/3)', [4, 0, 1, 4]]
K:(3 + -6n)/-2 = 1.311354802534919 : ['log_e(2)', [5, 4, 1, 5]]
Awesome. Several new constants, including a natural logarithm! And here I was ready to give up on this project for the day. The day is still young! *checks clock* Well, no, but it's still full of opportunity. It seem that when the degree of the denominator polynomial is larger than the degree of the numerator polynomial, we're more likely to get values that converge to Mobius transformations of famous constants, and I would probably focus my efforts on those ones if not for the fact that we just got our first logarithm at the end there when the numerator's degree was higher, so clearly I can't ignore those guys. They are rare gems. Also, all of the PCFs I've found related to sqrt(2/(e pi)) / erfc(1/sqrt(2)) had constants for the denominators. So we're getting different constants depending on the degree of the numerator versus the degree of the denominator. Awesome, right?
I guess I've got to look at 7 and 8 now. And then maybe try my hand at higher degrees again.
Ooh, first: This post isn't really mathematics. I've just been looking for numerical coincidences, not reasoning about things. But I'd like to register a prediction. The PCF "K:1/(5 + 10n)" will converge to a Mobius transform of {e^(2/5)}, possibly with Mobius coefficients of [1, 1, 1, -1]. Let's check that before looking at PCFs with coefficients in the 7s and 8s.
It converges to 5.066489563439472713631787788264925655136608101442522229..., which is .... ['e^(2/5)', [1, 1, 1, -1]]! Hell yes. I'd bet that the transformed constant of "K:1/(5 - 10n)" will be the same. There: now this post contains some mathematics; I have done some reasoning from the form of the polynomials to the form of the values. I still have no proofs, but there were starting to be a lot of numbers in this post for it to contain no mathematics. Here's the pattern I saw:
K:1/(1 + 2n) = 1.313035285499331 : ['e^(2/1)', [1, 1, 1, -1]]
K:1/(2 + 4n) = 2.163953413738653 : ['e^(2/2)', [1, 1, 1, -1]]
K:1/(3 + 6n) = 3.110296679619444 : ['e^(2/3)', [1, 1, 1, -1]]
K:1/(5 + 10n) = 5.066489563439473 : ['e^(2/5)', [1, 1, 1, -1]]
Or more generally,
K:(1/p(1 + 2n)) = ['e^(2/p)', [1, 1, 1, -1]]
And here's the related pattern for negative n:
K:1/(1 + -2n) = 0.23840584404423504 : ['e^(2/1)', [0, 2, 1, 1]]
K:1/(2 + -4n) = 1.5378828427399902 : ['e^(2/2)', [1, 3, 1, 1]]
K:1/(3 + -6n) = 2.6784872624683658 : ['e^(2/3)', [2, 4, 1, 1]]
K:1/(4 + -8n) = 3.755081337596291 : ['e^(2/4)', [3, 5, 1, 1]]
K:1/(5 + -10n) = 4.802624679775096 : ['e^(2/5)', [4, 6, 1, 1]]
Ore more generally,
K:1/(p(1 + -2n)) = ['e^(2/p)', [p-1, p+1, 1, 1]]
And here are some more families:
K:1/(3 + 2n) = 3.194528049465325 : ['e^(2/1)', [1, -1, 0, 2]]
K:1/(6 + 4n) = 6.09929355660769 : ['e^(2/2)', [1, -1, -1, 3]]
K:1/(9 + 6n) = 9.066456066042033 : ['e^(2/3)', [1, -1, -2, 4]]
K:1/(12 + 8n) = 12.04991096155901 : ['e^(2/4)', [1, -1, -3, 5]]
K:1/(15 + 10n) = 15.039954366828226 : ['e^(2/5)', [1, -1, -4, 6]]
K:4/(1 + 2n) = 2.0746294414550963 : ['e^(4/1)', [2, 2, 1, -1]]
K:4/(2 + 4n) = 2.6260705709986625 : ['e^(4/2)', [2, 2, 1, -1]]
K:4/(3 + 6n) = 3.431809417151078 : ['e^(4/3)', [2, 2, 1, -1]]
K:4/(4 + 8n) = 4.327906827477306 : ['e^(4/4)', [2, 2, 1, -1]]
K:4/(5 + 10n) = 5.263864883664377 : ['e^(4/5)', [2, 2, 1, -1]]
K:4/(6 + 12n) = 6.220593359238888 : ['e^(4/6)', [2, 2, 1, -1]]
K:4/(3 + 2n) = 3.722213300413417 : ['e^(4/1)', [4, -4, 1, 3]]
K:4/(6 + 4n) = 6.38905609893065 : ['e^(4/2)', [1, -1, 0, 1]]
K:4/(9 + 6n) = 9.26334591401584 : ['e^(4/3)', [4, -4, -1, 5]]
K:4/(12 + 8n) = 12.19858711321538 : ['e^(4/4)', [2, -2, -1, 3]]
K:4/(2 + -4n) = 0.4768116880884701 : ['e^(2/1)', [0, 4, 1, 1]]
K:4/(4 + -8n) = 3.0757656854799804 : ['e^(2/2)', [2, 6, 1, 1]]
K:4/(6 + -12n) = 5.3569745249367315 : ['e^(2/3)', [4, 8, 1, 1]]
K:4/(8 + -16n) = 7.510162675192582 : ['e^(2/4)', [6, 10, 1, 1]]
K:4/(10 + -20n) = 9.605249359550193 : ['e^(2/5)', [8, 12, 1, 1]]
To summarize the constants we've seen so far, we have {e} to various fractional powers, and the tangent of various fractions, we've got the two error function things that I could not have come up with myself and if there are more values like them, then it will take a small miracle for me to find them (sqrt(2/(e pi)) / erfc(1/sqrt(2)) and sqrt(pi * e / 2) * erf(sqrt(1/2))), and also we've seen log_e(2) and sqrt(2 pi). Also, I mentioned that pi and the Euler-Gompertz constant showed up for the quadratics, but I didn't specify which PCFs. I'm quite happy with how many constants have shown so far. This is a good project. I hope these aren't all just trivial cases of Gauss's continued fraction series, but even if they are, I'm having fun, and I'm okay with rediscovering and stating things specifically that Euler and Gauss might known generally. Also, who knows?, maybe there will be a new formula somewhere. It's possibly that Euler and Gauss didn't describe every possibly convergent PCF, isn't it? I think so. Not likely, but possible. Okay, let's look at some (1st/0th)s and (0th/1st)s with even higher coefficients, and then start on the quadratics. Tomorrow. Goodnight.
WAIT! I can get a fake polynomial for the cube root of two if I allow non-integer coefficients! It's the same trick as {K:1/(5 - 10n) = e^(2/5)}, but instead of putting 5 in the denominator of the power, I have to put {6/log_e(2)! The log_e(2) term turns the base of e^(2) into a 2 and the {6} term turns the power into 1/3! So with any luck, K:1/(6/ln(2) + 12n/ln(2)) = 2^(1/3) [1, 1, 1, -1]. ...testing... It does! It's not a PCF, I think because it doesn't have integer coefficients! That's part of the definition right, I honestly can't remember. There might not be an integer-coefficient PCF for 2^(1/3)! But it's by far the closest thing I've come up with so far. Good job, me. Tomorrow I'll try to find one that gives 2^(1/3) directly, instead of (2^(1/3) + 1) / (2^(1/3) - 1).
Oof. For the (1st/0th) PCFs with 7s and 8s in the coefficients, among the convergent ones that had irrational values, I tested 563 of them found no alternative expressions. I think I need to get more creative with my constants. ... Here we go: 4/563:
K:(8 + 8n)/4 = 6.260099161363913 : ['sqrt(pi) * e * erfc(1)', [2, 0, -1, 1]]
K:8n/4 = 5.2779350284695825 : ['sqrt(pi) * e * erfc(1)', [0, 4, 1, 0]]
K:8n/8 = 8.83632161198883 : ['sqrt(pi) * e^4 * erfc(2)', [0, 4, 1, 0]]
K:(8 + 8n)/8 = 9.56569803448636 : ['sqrt(pi) * e^4 * erfc(2)', [2, 0, -2, 1]]
The power of e is the square of the argument of erfc, so {sqrt(pi) * e^9 * erfc(3)} will probably show up eventually (for PCFs with larger coefficients).
Oh, cool, getting creative with constants also solved one of the 6 mystery PCFs, the one I'd called T6:
K:2n/2 = 2.6389675142347913 : ['sqrt(pi) * e * erfc(1)', [0, 2, 1, 0]]
Great success. T4 and T5 were also (linear/constant) PCFs. I bet if I keep getting creative in similar directions, those will also submit their secrets in time. Here are the (0th/1st)s with coefficients up to 8:
K:1/(4 + 8n) = 4.082988165073597 : ['sqrt(e)', [1, 1, 1, -1]]
K:-1/(4 + -8n) = 4.255341921221036 : ['tan(1/4)', [1, 4, 0, 1]]
K:-1/(4 + 8n) = 3.91631736464594 : ['tan(1/4)', [0, 1, 1, 0]]
K:-4/(4 + 8n) = 3.660975443424904 : ['tan(1/2)', [0, 2, 1, 0]]
K:8/(4 + -5n) = 0.3548238549860245 : ['sqrt(2 pi)', [0, 8, 7, 5]]
K:4/(4 + 8n) = 4.327906827477306 : ['e', [2, 2, 1, -1]]
K:-4/(4 + -8n) = 5.092604979687581 : ['tan(1/2)', [2, 4, 0, 1]]
K:4/(4 + -8n) = 3.0757656854799804 : ['e', [2, 6, 1, 1]]
K:4/(-7 + 7n) = 0.2819046589413371 : ['sqrt(pi /4)', [3, 0, 5, 5]]
K:1/(4 + -8n) = 3.755081337596291 : ['sqrt(e)', [3, 5, 1, 1]]
K:8/(-6 + 2n) = 0.9140696882921662 : ['sqrt(2 pi)', [4, 6, 5, 5]]
Smashing. On to the quadratics now?
Hm, I got too many new constants out from the quadratics at once, so I tested, and I've got some false positives. For everything above, I'd just tested that the alternative expression agreed with a difference of less than 0.00000001, but that's not cutting the mustard anymore. Like K:-2/(2 + 2n + -3n^2) matches ['sin(2)/cos(1/2)', [3, 1, 3, 5]] to that precision, and K:-4/(-3 + 3n + -2n^2) matches ['zeta(7)', [3, 0, 2, 4]] to that precision, but they're both unequal.
I guess I'll have to recheck the things above. I think most will be ok. It's not a coincidence that simple continued fractions are producing simple numbers like tan(1), but I'll recheck them all.
Some (0th/2nd) quadratics that I haven't ruled out yet:
K:4/(2 + n + -n^2) = 0.32731942245535756 : ['tan(2)', [0, -2, 6, 7]]
K:-1/(-n + -n^2) = 0.546156822289237 : ['tan(1/4)', [2, 1, 3, 2]]
K:3/(-1 + n + 2n^2) = 0.288719715477911 : ['e^3', [2, 1, 7, 2]]
K:-1/(2 + 3n + n^2) = 1.8309759380253865 : ['tan(1/4)', [3, 2, 2, 1]]
K:-4/(-2n + -2n^2) = 1.092313644578474 : ['tan(1/4)', [4, 2, 3, 2]]
K:-1/(2 + -3n + n^2) = 1.453843177710763 : ['tan(1/4)', [4, 3, 3, 2]]
K:4/(1 + 3n^2) = 1.9292908552521335 : ['tan(5)', [5, 0, 2, -2]]
K:-1/(3 + n + n^2) = 2.795419745554796 : ['tan(1/3)', [5, 3, 2, 1]]
K:-1/(3 + -3n + -n^2) = 4.168544394862332 : ['tan(1/5)', [5, 4, 1, 1]]
K:-4/(-1 + -2n + -n^2) = 0.12907450567234416 : ['cos(1)', [6, -2, 3, 8]]
K:-4/(-2n + 2n^2) = 3.661951876050773 : ['tan(1/4)', [6, 4, 2, 1]]
K:4/(3 + 3n + -3n^2) = 5.253889243859674 : ['sqrt(4 pi)', [6, 5, 0, 5]]
K:4/(-2 + n + n^2) = 2.3913711043812507 : ['tan(2)', [6, 7, 3, 4]]
K:-4/(3 + -n + 3n^2) = 2.146894061209078 : ['(1/(3e) + 2/3 * sqrt(e) * sec(sqrt(3)/2))', [7, -2, 0, 5]]
That last one looks a lot more complicated than the others, doesn't it? I don't have high hopes for it. I also had an expression for the reciprocal Fibonacci constant. That would have been cool.
And here come some conjectural (1st/2nd)s:
K:(-1 + -2n)/(1 + 3n + 2n^2) = 0.4700559958167652 : ['tan(1/2)', [-1, 2, 2, 2]]
K:(-1 + -n)/(2 + -2n + n^2) = 3.170278287105067 : ['e', [8, 5, 2, 3]]
K:(-2 + -n)/(-1 + -n^2) = 1.7089952210832067 : ['e', [4, 6, 4, -1]]
K:(-2 + -n)/(2 + 2n + -n^2) = 1.027508717545395 : ['e^(3/5)', [2, 3, 3, 1]]
K:(-2 + 3n)/(-n + 2n^2) = 0.6174416631274331 : ['pi * (e^pi + 1) / (e^pi - 1)', [5, 2, 7, 7]]
K:(-2n)/(1 + 3n + -2n^2) = 0.905946261815298 : ['pi/e', [0, 3, 2, 1]]
K:(-3 + -3n)/(3 + n + -n^2) = 0.3275387336077805 : ['sin(1)/cos(1/2)', [3, 1, 4, 8]]
K:(-3 + 2n)/(-1 + -3n + 3n^2) = 0.2395206759057082 : ['tan(2)', [2, 5, 2, 7]]
K:(-4 + -3n)/(1 + -n + n^2) = 1.5339264997648976 : ['zeta(5)', [7, 7, 8, 1]]
K:(-4 + -n)/(3 + -2n + 3n^2) = 1.5463295342131085 : ['acosh(2)', [7, 3, 6, 0]]
K:(-5 + -n)/(-1 + -2n + -2n^2) = 0.3489737590255696 : ['catalan', [0, 5, 8, 7]]
K:(-5 + 2n)/(1 + 2n + n^2) = 0.2287222870236921 : ['tan(2e)', [5, 8, -2, 8]]
K:(-6 + -3n)/(-1 + n + n^2) = 2.709080407813754 : ['sqrt(pi) * e^4 * erfc(2)', [1, 2, 2, 0]]
K:(-6 + -n)/(-3 + 2n + n^2) = 0.6911579406066546 : ['catalan', [4, 1, 3, 4]]
K:(-6 + n)/(1 + 3n + -2n^2) = 0.4084026684656323 : ['sin(1)/cos(1/2)', [0, 4, 5, 5]]
K:(-6 + n)/(3 + -2n^2) = 0.2729660552817177 : ['sqrt(3 e)', [-1, 5, 1, 5]]
K:(1 + -2n)/(1 + -n + -2n^2) = 1.6034844952968752 : ['cos(1)', [3, 8, 0, 6]]
K:(1 + -2n)/(n + -2n^2) = 2.127406115227631 : ['tan(1/2)', [2, 2, -1, 2]]
K:(1 + -3n)/(1 + 3n^2) = 0.4454718406840345 : ['sqrt(pi) * e^4 * erfc(2)', [7, 1, 3, 8]]
K:(1 + 3n)/(2n^2) = 1.4187693582820886 : ['sqrt(2 e)', [5, 3, 1, 8]]
K:(1 + n)/(2 + n + -n^2) = 0.5327158967264364 : ['tan(5)', [3, 2, 6, 5]]
K:(2 + n)/(n + n^2) = 1.143138287195733 : ['atan(1)', [4, 2, 7, -1]]
K:(3 + -2n)/(1 + -n + -2n^2) = 0.4700559958167652 : ['tan(1/2)', [-1, 2, 2, 2]]
K:(3 + -2n)/(3n + -2n^2) = 0.6236418268670918 : ['cos(1)', [0, 6, 3, 8]]
K:(3 + 3n)/(n + 2n^2) = 1.5575354707761788 : ['Euler-Gompertz constant', [7, 7, 7, 3]]
K:(4 + -3n)/(2n^2) = 0.5744237855650017 : ['e', [1, 5, 2, 8]]
K:(4 + n)/(-1 + 2n + 3n^2) = 0.13784384002114775 : ['pi', [-1, 7, 7, 6]]
K:(5 + -2n)/(1 + n + 2n^2) = 1.7332654683413418 : ['atan(3)', [4, 8, 2, 5]]
K:(5 + n)/(-2 + 2n + n^2) = 0.9110505744077929 : ['log_e(2/3)', [3, -1, 6, 0]]
K:(6 + 3n)/(-3 + 3n + n^2) = 0.5878653932612207 : ['1/(sqt(2) * csc(sqrt(2)) - 1)', [2, 1, 5, -2]]
K:n/(1 + -2n + -3n^2) = 0.7580161412280335 : ['e^(1/3)', [7, 4, 8, 7]]
K:n/(3n + -n^2) = 1.8771731914610505 : ['tan(5)', [6, 5, 3, 2]]
Even if some of those are wrong, I think we've seen enough of the space of convergent PCF values to guess that we're not going to get algebraic numbers of degree higher than two. This isn't too surprising: we have some results that canonical continued fractions won't converge to cubic algebraic numbers if their terms are too simple, e.g. "Automatic continued fractions are transcendental or quadratic", (Bugeaud, 2013). There are however generalized continued fractions where the odd terms of the numerator and denominator sequences are given by a pair polynomials and the even terms are given by a different pair of polynomials, and these can represent cube roots. So maybe in between PCFs with integer coefficients (which don't seem to be able to represent cubic irrationals) and PCFs with transcendental coefficients (which can represent cube roots, if you believe the formula I found), there's still the question of whether PCFs with rational or quadratic irrational coefficients can be used to generate cube roots. Ooh, or complex coefficients would be cool. I'll probably try rational coefficient PCFs when I get sick of these. But I think I'll stick with integer coefficients till the end of the month. There's lots more to play with when it comes to the quadratic polynomials and higher. And I need to check all these results to higher precision. Lots to do. We actually got very close to a cube root on that run: K:(5 + -2n)/(2n + 3n^2) = 0.5925786880061988 : ['3^1/3', [8, -2, 7, 6]], but it's not real.
I'm thinking back to those 6 mysterious PCFs that I wanted to solve. Let's just talk about T1 for specificity. Suppose it's a mobius transformation of some famous constant, and the mobius transformation has small coefficients, like in the range [-2, 2]. From [-2, -2, -2, -2] and [2, 2, 2, 2] there are 5^4 sequences = 625 sequences, but 25 of them have zero for the denominator of (az + b)/ (cz + d). Of the six hundred, another 104 are degenerate (they have a determinant (a*d - b*c) of zero and so just produce some rational number no matter what constant you input for transformation). Of the remaining transforms, 248 will produce negative numbers from positive constants, so if we assume the original untransformed constant is positive, and we see that T1 is positive, then we can disregard those 248 transformations. Of the remaining 248, lots of them will produce the same decimal part as another, and will only differ in the leading integer. There are just 72 different decimal parts. If I want to find the supposed original constant that generated T1 by a Mobius transformation with small coefficients by googling, then I only have 72 sequences to google, or 71 if we've already searched for the decimal part of T1.
On the other hand, if we look at the distribution of mobius transforms that showed up in the results above, the three most common results ([1 0 0 1], [1 1 0 1], [1 2 0 1]) - accounting for 19% of observed small transformations - they don't change the decimal part, and the fourth most common one ([1 1 0 2)], only changes the first number after the decimal relative to constant/2. So.....so maybe looking up 72 sequences isn't the smartest way to go. Maybe it's smarter to guess that the tail is correct, and look for different values of the ... I hate this. I don't want to google numbers. I want to find expressions by my own ingenuity. Let's do some more quadratics and then check all the results so far to some very high precision like 300 digits.
Here's (2nd/0th):
K:(1 + 2n + n^2)/5 = 5.637013560913436 : ['log_e(1/4)', [-1, -1, 5, 7]]
K:(n^2)/5 = 5.177398899134019 : ['log_e(2)', [0, 2, 2, -1]]
K:(n + n^2)/4 = 4.397319302471208 : ['log_e(1/4)', [0, 1, 2, 3]]
K:(n + n^2)/6 = 6.29393626341169 : ['log_e(1/4)', [0, -1, 3, 4]]
And (2nd/1st):
K:(n + n^2)/(2 + 2n) = 2.414213562373095 : ['sqrt(2):', [1, 1, 0, 1]]
K:(2n + 2n^2)/(2 + 2n) = 2.732050807568877 : ['sqrt(3):', [1, 1, 0, 1]]
K:(n + n^2)/(1 + n) = 1.618033988749895 : ['sqrt(5):', [1, 1, 0, 2]]
K:(-1 + n + 2n^2)/(1 + -2n) = 0.2465636898691922 : ['sin(1)/cos(2)', [1, 4, -1, 6]]
K:(-n^2)/(2 + 2n) = 1.6768750281787008 : ['Euler-Gompertz constant', [0, 1, 1, 0]]
K:(n^2)/(1 + 2n) = 1.2732395447351628 : ['atan(1)', [0, 1, 1, 0]]
K:(2n^2)/(2 + n) = 2.4663034623764317 : ['log_e(2/3)', [0, 1, -1, 0]]
Usually I remove the quadratic irrational numbers, but maybe it's interesting to see when they slip through? They're rare, but possible, I think. Unless those are false positives. That Euler-Gompertz is not a false positive, and I don't think atan(1) is either. We're just getting some cool numbers. Rare gems.
The (2nd/2nd) set is quite a large. I'll do it tomorrow. Or maybe I'll increase the coefficient size on (2nd/1st) first.
Ooh, or (2nd/2nd) with positive coefficients:
K:(2n^2)/(n + n^2) = 0.6471115953314572 : ['apery', [-1, 6, 7, -1]]
K:(2n^2)/(2n + n^2) = 0.5156696120811021 : ['log_e(5/6)', [0, 3, 1, 6]]
K:(2 + n^2)/(2n + 2n^2) = 0.6693547763230082 : ['tan(e)', [3, 5, -1, 5]]
K:(1 + n + n^2)/(2n^2) = 1.0695761060214275 : ['apery', [6, -1, 4, 1]]
K:(2 + 2n^2)/(n + n^2) = 1.1983144767768255 : ['reciprocal Fibonacci constant', [7, 5, 5, 7]]
And (2nd/2nd) with one negative coefficient out of six:
K:(n + n^2)/(1 + -n + -2n^2) = 0.23840584404423512 : ['e^2', [0, 2, 1, 1]]
K:(n + n^2)/(-1 + n^2) = 0.43312742672231175 : ['e', [1, 8, 8, 3]]
K:(-2 + -2n + n^2)/(1 + 2n + 2n^2) = 0.3810123103325837 : ['sqrt(2 pi)', [2, -1, 5, -2]]
K:(-1 + n + n^2)/(n^2) = 0.504361037223575 : ['sqrt(pi /3)', [2, 2, 1, 7]]
K:(-1 + -n + n^2)/(1 + 2n + n^2) = 0.7565415558353624 : ['sqrt(pi /3)', [3, 3, 1, 7]]
K:(-1 + -n + n^2)/(2 + 2n + n^2) = 1.8038142066648715 : ['sqrt(pi) * e * erfc(1)', [4, 2, 5, -1]]
K:(-2 + n + 2n^2)/(2 + -2n^2) = 1.111061934757286 : ['sqrt(pi * e / 2)', [5, 1, 3, 4]]
K:(2 + n + 2n^2)/(-2 + n^2) = 0.961091930726706 : ['sqrt(pi /4)', [5, 4, 2, 7]]
K:(2 + n + -2n^2)/(1 + 2n^2) = 1.397263491897355 : ['sqrt(5):', [6, 5, 5, 2]]
Well done. If I find a family of PCFs that converge to transformations of e^2, I should probably check if the coefficient patterns are simpler if I use tanh(1) = (e^2 - 1 )/(e^2 + 1) as my constant.
Here are the rest:
K:(-n + -n^2)/(1 + -n + -2n^2) = 2.5574077246549023 : ['tan(1)', [1, 1, 0, 1]]
K:(-1 + -n + -2n^2)/(2 + -2n^2) = 0.3618266154305907 : ['atan(3)', [1, 3, 7, 3]]
K:(-1 + 2n + -2n^2)/(1 + -2n + -2n^2) = 1.3968137539646355 : ['e^(4/5)', [2, 5, 6, -2]]
K:(-1 + -2n + -n^2)/(-1 + -n + -2n^2) = 0.2812760988521658 : ['log_e(1/6)', [6, 7, 8, 1]]
K:(1 + -2n^2)/(2 + -2n + -n^2) = 0.17788275452301663 : ['tan(1/5)', [0, 1, 8, 4]]
K:(-1 + -n + -2n^2)/(2 + n + n^2) = 0.20086682421462954 : ['sin(1)/cos(2)', [1, 1, 4, 3]]
K:(-1 + -n + n^2)/(2 + -2n + n^2) = 1.260825572607849 : ['sqrt(pi) * e * erfc(1)', [2, 8, 6, 3]]
I'm confused by the lack of pi formulas. I thought I'd gotten some pi formulas earlier when I was playing with quadratics. And I I know that other people have come up with quadratic PCFs for pi. Something's wrong here. The only one that showed up was {K:(4 + n)/(-1 + 2n + 3n^2) = 0.13784384002114775 : ['pi', [-1, 7, 7, 6]]} which looks to be a false positive.
So I'll address that. But first, I was looking for patterns in the PCFs that converge to rational powers of e, and generated a list of things I wanted to test. First, the new PCFs:
K:(3 + n)/(3 + n) = 3.8244701674557673 : ['e', [3, -6, -2, 6]]
K:(1 + n)/(3 + n) = 3.4409585712908757 : ['e', [2, -5, -4, 11]]
I think these didn't show up before because I didn't go large enough with my Mobius coefficients. I've got to make a cut off somewhere, and I liked to use (-5, 5) a lot in the past, and (-2, 9) more recently, but neither of those ranges would have worked here.
Those two results gave me confidence in two of the patterns I was investigating. Here ae some more terms of the first pattern:
K:(4 + n)/(4 + n) = 4.85160064959488 : ['e', [8, -24, -9, 24]]
K:(5 + n)/(5 + n) = 5.871296601732962 : ['e', [45, -120, -44, 120]]
K:(6 + n)/(6 + n) = 6.886288765578019 : ['e', [264, -720, -265, 720]]
Those numbers are factorials and subfactorials. That is fucking awesome. Holy shit.
Here are some terms of the second pattern:
K:(1 + n)/(1 + n) = 1.7182818284590453 : ['e', [1, -1, 0, 1]]
K:(1 + n)/(2 + n) = 2.549646778303845 : ['e', [1, -2, -1, 3]]
K:(1 + n)/(3 + n) = 3.4409585712908757 : ['e', [2, -5, -4, 11]]
K:(1 + n)/(4 + n) = 4.366328586643861 : ['e', [6, -16, -18, 49]]
K:(1 + n)/(5 + n) = 5.312419837301227 : ['e', [24, -65, -96, 261]]
K:(1 + n)/(6 + n) = 6.271879369490313 : ['e', [120, -326, -600, 1631]]
K:(1 + n)/(7 + n) = 7.240392798374601 : ['e', [720, -1957, -4320, 11743]]
The first Mobius coefficients are n!: https://oeis.org/A000142
The second Mobius coefficients are -Sum_(k=0, n) n!/k!: https://oeis.org/A000522
The third Mobius coefficients are -n(n!): https://oeis.org/A001563
The fourth Mobius coefficients are Sum_(k=0, n) (k + 1)n!/(n - k)!: https://oeis.org/A001339
Oh, ha, some {pi}s were hiding in arctans. atan(1) = pi/4, so
K:(n^2)/(1 + 2n) = 1.2732395447351628 : ['atan(1)', [0, 1, 1, 0]]
is 4/pi.
K:(n^2)/(1 + 2n) = 1.2732395447351628 : ['pi', [0, 4, 1, 0]]
Good, good. And
K:(2 + n)/(n + n^2) = 1.143138287195733 : ['atan(1)', [4, 2, 7, -1]]
after a little rearranging is just (4pi + 8) / (7pi - 4). Except it's not! False positive. I really need to check all my results to higher precision.
-
It was a long day. I just want to find a new (to me) formula. Let's look for the pattern behind (0th/1st) PCFs that converge to tangents of rational numbers. First up, I thought these polynomials should also converge to {tan()}s, so I looked for Mobius coefficients and eventually found expression for all of them:
K:-1/(5 + -2n) = 4.616708305766208 : ['tan(1)', [14, 9, 3, 2]]
K:-1/(6 + -4n) = 5.60727368253041 : ['tan(1/2)', [6, 11, 1, 2]]
K:-4/(3 + -2n) = 4.186915540460184 : ['tan(2)', [6, -1, 2, 1]]
K:-4/(6 + -4n) = 5.217958098460862 : ['tan(1)', [6, 4, 1, 1]]
K:2/(-5 + -n) = -5.318668524094481 : ['tan(1/3)', [9, 20, -1, -4]]
And that led to more guesses:
K:-4/(5 + -10n) = 5.845586437476324 : ['tan(2/5)', [2, 5, 0, 1]]
K:-4/(6 + -12n) = 6.692507099021151 : ['tan(1/3)', [2, 6, 0, 1]]
K:-4/(7 + -14n) = 7.587502723239725 : ['tan(2/7)', [2, 7, 0, 1]]
K:-1/(5 + -10n) = 5.202710035508672 : ['tan(1/5)', [1, 5, 0, 1]]
K:-1/(5 + 10n) = 4.933154875586894 : ['tan(1/5)', [0, 1, 1, 0]]
K:-4/(6 + 12n) = 5.776114072554554 : ['tan(1/3)', [0, 2, 1, 0]]
K:-4/(5 + 10n) = 4.7304448400782215 : ['tan(2/5)', [0, 2, 1, 0]]
K:-4/(7 + 14n) = 6.808479078943505 : ['tan(2/7)', [0, 2, 1, 0]]
K:-1/(7 + 2n) = 6.887746993529785 : ['tan(1)', [-2, 3, -9, 14]]
K:-4/(9 + 6n) = 8.72987874083479 : ['tan(2/3)', [4, 0, 3, -2]]
K:-1/(9 + 2n) = 8.908447367646522 : ['tan(1)', [-9, 14, -61, 95]
K:-1/(11 + 2n) = 10.922678838321968 : ['tan(1)', [-61, 95, -540, 841]]
And one more guess that I haven't found an expression for:
6.7833954554263233037: "K:-1/(7 + -2n)
even when I went up to (-40, 40) for Mobius coefficients, but I still think it exists and maybe I can find it by reasoning rather than brute force search. Let's talk about the PCFs positive n in the denominator first.
From...
K:-1/(1 + 2n) : ['tan(1)', [0, 1, 1, 0]]
K:-1/(2 + 4n) : ['tan(1/2)', [0, 1, 1, 0]]
K:-1/(3 + 6n) : ['tan(1/3)', [0, 1, 1, 0]]
K:-1/(4 + 8n) : ['tan(1/4)', [0, 1, 1, 0]]
K:-1/(5 + 10n) : ['tan(1/5)', [0, 1, 1, 0]]
we induce that
K:-1/(c * (1 + 2n)) = ['tan(1/c)', [0, 1, 1, 0].
Nice.
Another pattern:
K:-4/(2 + 4n) : ['tan(1)', [0, 2, 1, 0]]
K:-4/(3 + 6n) : ['tan(2/3)', [0, 2, 1, 0]]
K:-4/(4 + 8n) : ['tan(1/2)', [0, 2, 1, 0]]
K:-4/(5 + 10n) : ['tan(2/5)', [0, 2, 1, 0]]
K:-4/(6 + 12n) : ['tan(2/6)', [0, 2, 1, 0]]
K:-4/(7 + 14n) : ['tan(2/7)', [0, 2, 1, 0]]
from which we induce
K:-4/(c * (1 + 2n)) : ['tan(2/c)', [0, 2, 1, 0]]
Nice. I wonder what constant we need on top to get tan(3/c) or tan(4/c)! We'll find one before the night is through, I should think. How about a trickier pattern?
K:-1/(3 + 2n) : ['tan(1)', [-1, 0, -1, 1]]
K:-1/(5 + 2n) : ['tan(1)', [-1, 1, -2, 3]]
K:-1/(7 + 2n) : ['tan(1)', [-2, 3, -9, 14]]
K:-1/(9 + 2n) : ['tan(1)', [-9, 14, -61, 95]
K:-1/(11 + 2n) : ['tan(1)', [-61, 95, -540, 841]]
The first and third Mobius coefficients come from: https://oeis.org/A053983
The second and fourth Mobius coefficients come from the closely related: https://oeis.org/A053984
Those sequences are both about continued fraction approximations of tan(1), and there is some cool stuff in the links. More weird Bessel functions for example.
A few more (0th/1st) with positive n that I don't see patterns for yet:
K:-4/(-1 + 2n) : ['tan(2)', [2, 1, 0, -1]]
K:-4/(3 + 2n) : ['tan(2)', [4, 0, 1, -2]]
K:-4/(6 + 4n) : ['tan(1)', [2, 0, 1, -1]]
K:-4/(9 + 6n) : ['tan(2/3)', [4, 0, 3, -2]]
And you might be like, "hey, dummy, {3, 6, 9, ...} and {2, 4, 6, ...} are already patterns, and I'd be like, oh yeah, you're right, how did I miss that, but also it's been a really bad day. I guess I should figure out Mobius coefficients for K:-4/(12 + 8n) now.
All of these positive {n} PCFs for tangents of rational numbers have buddies with negative {n} that converge to values involving the same constant, but with different Mobius coefficients. For example,
K:-4/(7 + -14n) : ['tan(2/7)', [2, 7, 0, 1]]
K:-4/(7 + 14n) : ['tan(2/7)', [0, 2, 1, 0]]
K:2/(-5 + -n) : ['tan(1/3)', [9, 20, -1, -4]]
K:2/(-5 + n) : ['tan(1/3)', [4, 0, 1, 4]]
K:-1/(2 + -4n) : ['tan(1/2)', [1, 2, 0, 1]]
K:-1/(2 + 4n) : ['tan(1/2)', [0, 1, 1, 0]]
So there are different but related patterns for the negative {n} PCFs. I'm not really in the mood to find them right now though. I think I'd rather find a rule that generalizes:
K:-1/(c * (1 + 2n)) = ['tan(1/c)', [0, 1, 1, 0].
K:-4/(c * (1 + 2n)) : ['tan(2/c)', [0, 2, 1, 0]]
So I'll play with different numerators of the PCF for a while.
...
Oh, it's beautiful! The -1 and -4? Negative perfect squares, -n^2. And the Mobius Coefficients? [0, n, 1, 0]. For example:
K:-9/(1 + 2n) = -21.0457576543036 : ['tan(3/1)', [0, 3, 1, 0]]
K:-9/(2 + 4n) = 0.21274453290795733 : ['tan(3/2)', [0, 3, 1, 0]]
K:-9/(3 + 6n) = 1.926277847802992 : ['tan(3/3)', [0, 3, 1, 0]]
K:-9/(4 + 8n) = 3.220278445648132 : ['tan(3/4)', [0, 3, 1, 0]]
K:-9/(5 + 10n) = 4.385087841234307 : ['tan(3/5)', [0, 3, 1, 0]]
K:-9/(6 + 12n) = 5.491463165137356 : ['tan(3/6)', [0, 3, 1, 0]]
K:-9/(7 + 14n) = 6.566087241562885 : ['tan(3/7)', [0, 3, 1, 0]]
K:-9/(8 + 16n) = 7.621436619073163 : ['tan(3/8)', [0, 3, 1, 0]]
K:-9/(9 + 18n) = 8.66417110883183 : ['tan(3/9)', [0, 3, 1, 0]]
K:-9/(10 + 20n) = 9.698184431297483 : ['tan(3/10)', [0, 3, 1, 0]]
K:-9/(11 + 22n) = 10.725910708907033 : ['tan(3/11)', [0, 3, 1, 0]]
K:-9/(12 + 24n) = 11.74895209393782 : ['tan(3/12)', [0, 3, 1, 0]]
K:(-16)/(1 + 2n) = 3.4547646178024665 : ['tan(4/1)', [0, 4, 1, 0]]
K:(-16)/(2 + 4n) = -1.830630217441143 : ['tan(4/2)', [0, 4, 1, 0]]
K:(-16)/(3 + 6n) = 0.9681177078207467 : ['tan(4/3)', [0, 4, 1, 0]]
K:(-16)/(4 + 8n) = 2.568370463737323 : ['tan(4/4)', [0, 4, 1, 0]]
K:(-16)/(5 + 10n) = 3.8848584026018975 : ['tan(4/5)', [0, 4, 1, 0]]
K:(-16)/(6 + 12n) = 5.083606973533403 : ['tan(4/6)', [0, 4, 1, 0]]
K:(-16)/(7 + 14n) = 6.2209763557037805 : ['tan(4/7)', [0, 4, 1, 0]]
K:(-16)/(8 + 16n) = 7.321950886849808 : ['tan(4/8)', [0, 4, 1, 0]]
K:(-16)/(9 + 18n) = 8.399453953418282 : ['tan(4/9)', [0, 4, 1, 0]]
K:(-16)/(10 + 20n) = 9.460889680156443 : ['tan(4/10)', [0, 4, 1, 0]]
K:(-16)/(11 + 22n) = 10.510822821017381 : ['tan(4/11)', [0, 4, 1, 0]]
K:(-16)/(12 + 24n) = 11.552228145109108 : ['tan(4/12)', [0, 4, 1, 0]]
K:(-16)/(13 + 26n) = 12.58713062944253 : ['tan(4/13)', [0, 4, 1, 0]]
K:(-16)/(14 + 28n) = 13.61695815788701 : ['tan(4/14)', [0, 4, 1, 0]]
So now we induce:
K:-(d^2)/(c * (1 + 2n)) = ['tan(d/c)', [0, d, 1, 0].
And that's where I call it a night. Good job. And there's plenty here almost done for tomorrow and the next night. We're in good shape for discovery.
Oh cool, the formula still works when the numerator isn't the negative of a perfect square.
K:-2/(1 + 2n) = 0.22326917525195267 : ['sqrt(2) * tan(3/2)', [0, 2, 1, 0]]
K:-2/(1 + -2n) = 9.957797231718436 : ['sqrt(2) * tan(3/2)', [1, 1, 0, 1]]
Oh, wait, that's ... correct but stupid. I should have used {sqrt(2) divided by tan(3/2)} for the transformed constant. It still works because 2/sqrt(2) = sqrt(2), but the Mobius coefficients are messed up. Oh well, you get the idea.
...
Ha! I was playing with double factorials (which are not at all like nested factorials), since they're related to those complicated constants that have an erf() and a sqrt() and a {pi} in them, and I found this:
K:(2 + 2n)/(2 + n) = ['Sum(k=0, inf) 2^(2k)/(2k)!!', [2, -10, -1, 9]]
I thought it might be a false positive, but no, it's right to 100+ decimal places. But you know what that stupid complicated sum reduces to? It's just e^2. So really, none of the hundreds of double factorial expressions I checked helped me find new erf() constants. Ah well.
Today I mainly worked on finding alternative expression for the larger members of this series:
K:n/1 = 1.5251352761651504277 : sqrt(2/(e pi)) / erfc(1/sqrt(2)) % [1, 0, 0, 1]
K:n/2 = 2.3732155328228408673
K:n/3 = 3.2830986549304365069
K:n/4 = 4.2256071444894710728
K:n/5 = 5.1865039671258421156
K:n/6 = 6.1584826045445989173
K:n/7 = 7.1375456132265032765
I tried looking for separate patterns in the numerators and denominators of convergents (which are the sequence of rational approximations to the true PCF value after evaluating a few terms). I tried graphing y = K:n/x for real x and then tried figuring out the equation of curve by looking at the asymptotes and key values. Maybe I should have tried looking for complex zeroes numerically or something but I haven't yet.
I did find a new new erf() constant though:
K:(-2 + -n)/(-2n) = 4.403417872407463 : ['e ** (5 / 2) * sqrt(pi / 2) * erf(5 / sqrt(2))', [13, -10, 3, -3]]
Although it's a little suspicious because those coefficients are big and I've never gotten an e^(3/2) sqrt(pi/2) erf() constant.
I think now is a good time to look at all the erf() and erfc() constants that have shown up and find some patterns. Some patterns that I'm still figuring out have led me to these:
K:(-1 + -2n)/(4 + 2n) = 3.434949504371802 : ['sqrt(e pi / 2) * erf(1 / sqrt(2))', [1, 0, 1, -1]]
K:(2 + 2n)/(-4 + 2n) = 0.7694768308048724 : ['sqrt(e pi / 2) * erf(1 / sqrt(2))', [2, 6, 6, 3]]
K:(2 + 2n)/(4 + 2n) = 4.598234921289543 : ['sqrt(e pi / 2) * erf(1 / sqrt(2))', [2, -2, -2, 3]]
which we'll get back to.
But first, how about this:
K:(0 + n)/1 = 1.5251352761651504 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [0, -1, -1, 0]]
K:(1 + n)/1 = 1.9042712333296918 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [1, 0, -1, 1]]
K:(2 + n)/1 = 2.2117257812521967 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [2, -2, -2, 1]]
K:(3 + n)/1 = 2.4758076839580085 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [6, -3, -4, 3]]
K:(4 + n)/1 = 2.7103802506252745 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [8, -6, -5, 3]]
K:(5 + n)/1 = 2.9233265519552654 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [25, -15, -13, 9]]
K:(6 + n)/1 = 3.1195950551969576 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [39, -27, -19, 12]]
K:(7 + n)/1 = 3.3025176127171285 : ['sqrt(e pi / 2) * erfc(1 / sqrt(2))', [133, -84, -58, 39]]
All the same constant, but I don't know what the pattern is that generates the Mobius coefficients! It's very exciting. The fourth Mobius coefficients are documented (https://oeis.org/A225436) and the other sequences seem to be new.
The OEIS link above mentioned {sqrt(2 / e pi) / erfc(1 / sqrt(2)) - 1}, so I tried that as the constant to transform, and I got different Mobius coefficients that are a bit related to the ones above:
K:(1 + n)/1 = 1.9042712333296918 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [0, -1, -1, 0]]
K:(2 + n)/1 = 2.2117257812521967 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [2, 0, -1, 1]]
K:(3 + n)/1 = 2.4758076839580085 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [3, -3, -3, 1]]
K:(4 + n)/1 = 2.7103802506252745 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [6, -2, -3, 2]]
K:(5 + n)/1 = 2.9233265519552654 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [15, -10, -9, 4]]
K:(6 + n)/1 = 3.1195950551969576 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [27, -12, -12, 7]]
K:(7 + n)/1 = 3.3025176127171285 : ['sqrt(2 / e pi)/erfc(1 / sqrt(2)) - 1', [84, -49, -39, 19]]
The first column in this series looks like the second column in the previous series. The third column in this series looks like the fourth column in the previous series. And the other two columns are new, and the second one isn't even increasing in absolute magnitude, which is a little interesting. I wonder if there's a different version of this erfc() constant that makes all the columns of coefficients algorithmically simple.
I think I'll leave it there for the night. Goodnight.
It seemed that lots of the erf() and erfc() constants were coming from PCFs that had all even coefficients, so I'm looking over those again. Here's what the new search has turned up so far:
K:(-2 + -4n)/(4 + 4n) = 3.1545322347409073 : ['e ** (3) * sqrt(pi /2) * erfc(1/sqrt(2))', [8, 18, 3, 2]]
K:(-2 + -4n)/(6 + -2n) = 0.2441716203881457 : ['sqrt(pi /2) * erfc(0)', [13, -10, 7, 17]]
K:(-2 + 4n)/(2 + -4n) = 1.3025910785144221 : ['e * sqrt(pi /2) * erfc(2/sqrt(2))', [13, 14, -11, 14]]
K:(-2 + 4n)/(4 + -2n) = 0.39034155041240587 : ['e * sqrt(pi /2) * erfc(2/sqrt(2))', [9, 6, 19, 16]]
K:(-6 + -4n)/(-2 + -4n) = 0.2891041446674012 : ['e ** (3) * sqrt(pi /2) * erfc(2/sqrt(2))', [17, -10, 19, 11]]
K:(-6 + 4n)/(-2n) = 0.8289753487627823 : ['e * sqrt(pi /2) * erfc(2/sqrt(2))', [11, 14, 19, 16]]
K:(-6 + 4n)/(4 + -2n) = 1.3597373239451467 : ['e * sqrt(pi /2) * erfc(2/sqrt(2))', [11, 14, 10, 10]]
K:(2 + -4n)/(-4n) = 0.6340084206380814 : ['e ** (3) * sqrt(pi /2) * erfc(1/sqrt(2))', [3, 2, 4, 9]]
K:(2 + -4n)/(-6 + 2n) = 0.2337287336551362 : ['e ** (1/2) * sqrt(pi /2) * erfc(3/sqrt(2))', [9, 3, 9, 13]]
K:(2 + -4n)/(4 + 4n) = 3.73230329982846 : ['e ** (3) * sqrt(pi /2) * erfc(1/sqrt(2))', [5, 16, 1, 7]]
K:(2 + -4n)/2n = 0.6402626760548533 : ['e * sqrt(pi /2) * erfc(3/sqrt(2))', [-10, 11, 4, 17]]
K:(2 + 2n)/(-2 + 2n) = 0.32644401980111964 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [2, 0, 4, 3]]
K:(2 + 2n)/(-4 + 2n) = 0.7694768308048724 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [2, 6, 6, 3]]
K:(2 + 2n)/(2 + 2n) = 2.821372269284896 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [2, 0, 0, 1]]
K:(2 + 2n)/(4 + 2n) = 4.598234921289543 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [2, -2, -2, 3]]
K:(2 + 2n)/(6 + 2n) = 6.467050171779698 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [6, -8, -12, 17]]
K:(2 + 2n)/2 = 3.1300495806819564 : ['erfc(1/sqrt(3))', [-8, 17, -16, 11]]
K:(2 + 2n)/2n = 1.2616860984829235 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [2, 2, 2, 1]]
K:(4 + 2n)/(-2 + 2n) = 0.8386664076372042 : ['e ** (3/2) * sqrt(pi /2) * erfc(3/sqrt(2))', [7, 11, 16, 13]]
K:(4 + 2n)/(-2 + 2n) = 0.8386664076372042 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [12, 6, 13, 9]]
K:(4 + 2n)/(2 + -4n) = 0.09842254838360667 : ['sqrt(pi /2) * erfc(1/sqrt(2))', [-2, 2, -17, 19]]
K:(4 + 2n)/(2 + 2n) = 3.1703606822724604 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [4, 2, 1, 1]]
?K:(4 + 2n)/(4 + -4n) = 0.1606008998317945 : ['e ** (3/2) * sqrt(pi /2) * erfc(2/sqrt(2))', [1, 2, 8, 12]]3]]
K:(4 + 2n)/(4 + 2n) = 4.869899008743604 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 2, 1, -1]]
K:(4 + 2n)/2n = 1.7193622395186399 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [8, 6, 5, 3]]
K:(4 + 4n)/(-6 + 4n) = 2.4325527764063435 : ['erf(sqrt(3))', [11, 18, 10, 2]]
K:(4 + 4n)/2 = 3.808542466687572 : ['e ** (1/2) * sqrt(pi /2) * erfc(1/sqrt(2))', [2, 0, -1, 1]]
K:(4 + 4n)/4 = 5.358833767911172 : ['e ** (2) * sqrt(pi /2) * erfc(2/sqrt(2))', [2, 0, -2, 1]]
K:(6 + -2n)/(2 + -4n) = 0.2857142857142857 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-2, 4, -7, 14]]
K:(6 + -2n)/4n = 0.9411764705882353 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 16, 0, 17]]
K:(6 + -4n)/4n = 0.5358621310577631 : ['e ** (3) * sqrt(pi /2) * erfc(1/sqrt(2))', [2, 14, 5, 16]]
K:(6 + 2n)/(-2 + 2n) = 1.2481519198745676 : ['e ** (1/2) * sqrt(pi /2) * erfc(0)', [11, -10, 3, 4]]
K:(6 + 2n)/(-8 + 2n) = 1.7191422041792745 : ['erfc(sqrt(1/4))', [15, 12, -8, 15]]
K:(6 + 2n)/(2 + 2n) = 3.4896660297016795 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [15, 9, 4, 3]]
K:(6 + 2n)/(4 + 2n) = 5.12662471568162 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [3, 3, 1, 0]]
K:(6 + 2n)/(6 + 2n) = 6.897352381934315 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [3, -3, -2, 3]]
K:(6 + 4n)/(-2 + 4n) = 0.5985175545953603 : ['e * sqrt(pi /2) * erfc(2/sqrt(2))', [15, -6, 12, -8]]
K:(6 + 4n)/(-8 + 2n) = 0.5900440213064015 : ['erfc(sqrt(3))', [11, 10, 15, 17]]
K:(8 + 2n)/(-6 + 2n) = 0.8231179081380211 : ['e ** (3) * sqrt(pi /2) * erfc(3/sqrt(2))', [5, 11, -18, 15]]
K:(8 + 2n)/(6 + 2n) = 7.100856113528374 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [8, 0, -1, 3]]
K:(8 + 4n)/(-2 + 2n) = 1.7142857142857142 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 12, 0, 7]]
K:(8 + 4n)/(-4 + 2n) = 1.2307692307692308 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 16, 0, 13]]
K:(8 + 4n)/2 = 4.423451562568677 : ['e ** (1/2) * sqrt(pi /2) * erfc(1/sqrt(2))', [-4, 4, 2, -1]]
K:(8 + 4n)/4 = 5.887401526897414 : ['e ** (2) * sqrt(pi /2) * erfc(2/sqrt(2))', [8, -4, -5, 2]]
K:-2/(4 + -4n) = 0.25537670923216504 : ['e ** (5/2) * sqrt(pi /2) * erfc(1/sqrt(2))', [4, -3, 12, 6]]
K:-4/(4 + -2n) = 0.7309003044877358 : ['sqrt(pi /2) * erfc(0)', [19, 4, 16, 18]]
K:-4/(4 + -4n) = 0.5222285285105925 : ['e * sqrt(pi /2) * erfc(3/sqrt(2))', [3, -1, 15, -2]]
K:-6/(2 + 4n) = 0.8831886101616455 : ['e ** (2) * sqrt(pi /2) * erfc(0)', [-10, 17, -10, 7]]
K:-6/(8 + -4n) = 0.5164515163177982 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [9, 3, 18, 5]]
K:2/4n = 0.47114816114189134 : ['sqrt(pi /2) * erfc(3/sqrt(2))', [-10, 9, 9, 19]]
K:2n/(-4 + 2n) = 2.1266247156816203 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 3, 1, 0]]
K:2n/(2 + -2n) = 0.41481965886376976 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, 1, 1]]
K:2n/(2 + 2n) = 2.434949504371802 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, 1, -1]]
K:2n/(4 + 2n) = 4.309260009877055 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, 3, -4]]
K:2n/(6 + -2n) = 3.400831422165108 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 15, 1, 3]]
K:2n/2n = 0.7088749052272068 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, 1, 0]]
K:4n/2 = 3.050270552330301 : ['e ** (1/2) * sqrt(pi /2) * erfc(1/sqrt(2))', [0, 2, 1, 0]]
K:4n/4 = 4.746431065645682 : ['e ** (4/2) * sqrt(pi /2) * erfc(2/sqrt(2))', [0, 2, 1, 0]]
K:6/(-8 + 4n) = 0.48511579476364886 : ['erfc(sqrt(3))', [-12, 6, 1, 12]]
K:8/(-6 + 2n) = 0.9140696882921662 : ['sqrt(pi /2) * erfc(0)', [8, 6, 10, 5]]
K:8/(-8 + 2n) = 0.7520679248724212 : ['e ** (3) * sqrt(pi /2) * erfc(0)', [4, 6, 5, 16]]
What a mess. I will unmuss it.
Usually I don't include PCFs that converge to negative values, since for each one a closely related PCF will converge to the same absolute value, but I think these ones below will help complete some patterns:
K:2n/(-2n) = -0.7088749052272068 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, -1, 1, 0]]
K:2n/(-2 + 2n) = -0.41481965886376976 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, -1, 1, 1]]
K:2n/(4 + -2n) = -2.1266247156816203 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 3, -1, 0]]
And this guy fills a hole in a pattern that I'm expecting, and I don't know why it didn't show up earlier:
K:(4 + 2n)/(6 + 2n) = 6.686336517062026 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-4, 6, 5, -7]]
And there are 19 other holes that I'm trying to fill, not all with the same level of confidence that alternative expressions will show up. Some of them just won't work out, and maybe a few need larger Mobius coefficients, and I expect that for some of them, I'll need to get more creative with the form of my constant library, but I've done it before, and I'll do it again. More hole fillers:
K:(2 + 2n)/(-6 + 2n) = -1.0440284505924005 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-2, 24, -8, -9]]
K:(4 + 2n)/(-4 + 2n) = 0.8071071353260958 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [16, 18, 25, 15]]
K:(6 + 2n)/(-4 + 2n) = 1.0032881765846773 : ['e * sqrt(pi /2) * erfc(3/sqrt(2))', [7, 28, -3, 28]]
K:(6 + 2n)/(-6 + 2n) = 0.49097235663490124 : ['erf(sqrt(2))', [19, -7, 29, -5]]
K:(6 + 2n)/(8 + 2n) = 8.742067267065968 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [15, -21, -17, 24]]
K:2n/(6 + 2n) = 6.238613764218355 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, -1, -15, 21]]
K:8/(-4 + 2n) = 1.1570609439396708 : ['sqrt(pi /2) * erfc(0)', [20, 10, 17, 9]]
K:8/(-2 + 2n) = 1.5512711769290246 : ['e * sqrt(pi /2) * erfc(3/sqrt(2))', [20, 42, 21, 27]]
K:2n/(-6 + 2n) = -3.400831422165108 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 15, -1, -3]]
K:(-3 + -2n)/(6 + 2n) = 5.309260009877055 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-3, 3, -3, 4]]
K:(-5 + -2n)/(8 + 2n) = 7.238613764218355 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-15, 20, -15, 21]]
K:(5 + -2n)/(-2 + 2n) = 3.1266247156816203 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [1, 3, 1, 0]]
K:(7 + -2n)/(-4 + 2n) = -2.400831422165108 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [-1, 12, -1, -3]]
K:2n/(-2 + -2n) = -2.434949504371802 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, -1, 1]]
K:2n/(-4 + -2n) = -4.309260009877055 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, -3, 4]]
K:2n/(-6 + -2n) = -6.238613764218355 : ['e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))', [0, 1, -15, 21]]
-
Okay, the erf() look like a manageable puzzle, but I need to do some work on putting the erfc() constants into a consistent format, because a lot of the ones in my constant dictionary are equivalent or Mobius-equivalent to each other, and that needs to be fixed. But let's look at just the erf() ones for now. Almost all of them are have the same constant: 'e ** (1 / 2) * sqrt(pi / 2) * erf(1 / sqrt(2))'. Let's call it Serf for short.
The only constants containing erf() that weren't Serf were:
* K:(-2 + -n)/(-2n) : ['e ** (5 / 2) * sqrt(pi / 2) * erf(5 / sqrt(2))', [13, -10, 3, -3]]
* K:(6 + 2n)/(-6 + 2n) : ['erf(sqrt(2))', [19, -7, 29, -5]]
* K:(4 + 4n)/(-6 + 4n) : ['erf(sqrt(3))', [11, 18, 10, 2]]
The first of those is Serf with some extra 5s in interesting places, and I wish I had other examples of Serf with other integers in those places, and I've searched but they haven't shown up yet. The other two make we want to search for more erf(sqrt(n)) for integer n, but you might notice that the constant between those two don't seem to be the start of simple series. I could be wrong, but they're not obviously anything right now.
I've got more than 35 PCFs that converge to Mobius transformations of Serf and I'm still trying to put them into families so that the coefficient make sense.
These seem to be a family.
K:(-9 + -2n)/(12 + 2n) : [Serf, [-945, 1332, -945, 1333]]
K:(-7 + -2n)/(10 + 2n) : [Serf, [-105, 147, -105, 148]]
K:(-5 + -2n)/(8 + 2n) : [Serf, [-15, 20, -15, 21]]
K:(-3 + -2n)/(6 + 2n) : [Serf, [-3, 3, -3, 4]]
K:(-1 + -2n)/(4 + 2n) : [Serf, [-1, 0, -1, 1]]
K:(1 + -2n)/(2 + 2n) : [Serf, [1, 1, 1, 0]]
K:(3 + -2n)/(0 + 2n) : [Serf, [1, 0, 1, 1]]
K:(5 + -2n)/(-2 + 2n) : [Serf, [1, 3, 1, 0]]
K:(7 + -2n)/(-4 + 2n) : [Serf, [1, -12, 1, 3]]
When the first number of the upper polynomial is negative, we're getting the double factorials in the first and third column! A001147 = [1, 1, 3, 15, 105, 945, 10395, ...].
The fourth column is also a doublel factorial thing, namely, A286286 = [0, 1, 4, 21, 148, 1333, 14664, 190633, ...], and in the link you'll find that Seiichi Manyama expresses the sequence as{a(n) = (2*n-1)!! * Sum_{k=1..n} 1/(2*k-1)!!}. Cool thing, Seiichi Manyama.
But what happens when we extend in the other direction? ...
Oh shit, I've got it!
K:(1 + -2n)/(2 + 2n) : [Serf, [1, 1, 1, 0]]
K:(3 + -2n)/(0 + 2n) : [Serf, [1, 0, 1, 1]]
K:(5 + -2n)/(-2 + 2n) : [Serf, [1, 3, 1, 0]]
K:(7 + -2n)/(-4 + 2n) : [Serf, [1, -12, 1, 3]]
K:(9 + -2n)/(-6 + 2n) : [Serf, [1, 93, 1, -12]]
K:(11 + -2n)/(-8 + 2n) : [Serf, [1, -852, 1, 93]]
K:(13 + -2n)/(-10 + 2n) : [Serf, [1, 9543, 1, -852]]
The second and fourth columns come from A263801 = [1, 0, 3, -12, 93, -852, 9543, -125592, 1901433, -32557992, ...], {a(n) = Sum_{k=0..n} (-1)^k*(2*k-1)!!}.
I've been calling the constant Serf after Srinivasa and erf(), but I'd kind of like a one-letter symbol. Ramanujan spoke Tamil and Kannada besides English, so instead of an S, it might be appropriate to use சீ or ಶ್ರೀ. I think the second one is prettier. But it actually takes up about the same amount of horizontal space on my screen as writing out Serf, so, ...
Here's another family:
K:2n/(-12 + -2n) : [Serf, [0, 1, -10395, 14664]]
K:2n/(-10 + -2n) : [Serf, [0, 1, -945, 1333]]
K:2n/(-8 + -2n) : [Serf, [0, 1, -105, 148]]
K:2n/(-6 + -2n) : [Serf, [0, 1, -15, 21]]
K:2n/(-4 + -2n) : [Serf, [0, 1, -3, 4]]
K:2n/(-2 + -2n) : [Serf, [0, 1, -1, 1]]
K:2n/(0 + -2n) : [Serf, [0, 1, -1, 0]]
K:2n/(2 + -2n) : [Serf, [0, 1, 1, 1]]
K:2n/(4 + -2n) : [Serf, [0, 3, -1, 0]]
K:2n/(6 + -2n) : [Serf, [0, 15, 1, 3]]
K:2n/(8 + -2n) : [Serf, [0, 105, -1, 12]]
K:2n/(10 + -2n) : [Serf, [0, 945, 1, 93]]
K:2n/(12 + -2n) : [Serf, [0, 10395, -1, 852]]
with familial double-factorial sequences for the Mobius coefficients.
Briefly, those were K:(x + -2n)/(y + 2n) and K:(0 + 2n)/(y + -2n) for odd x and even y. Here are some families that have positive {2n} on both the numerator and denominator:
K:2n/(-10 + 2n) : [Serf, [0, 945, -1, -93]]
K:2n/(-8 + 2n) : [Serf, [0, 105, 1, -12]]
K:2n/(-6 + 2n) : [Serf, [0, 15, -1, -3]]
K:2n/(-4 + 2n) : [Serf, [0, 3, 1, 0]]
K:2n/(-2 + 2n) : [Serf, [0, -1, 1, 1]]
K:2n/(0 + 2n) : [Serf, [0, 1, 1, 0]]
K:2n/(2 + 2n) : [Serf, [0, 1, 1, -1]]
K:2n/(4 + 2n) : [Serf, [0, 1, 3, -4]]
K:2n/(6 + 2n) : [Serf, [0, 1, 15, -21]]
K:2n/(8 + 2n) : [Serf, [0, 1, 105, -148]]
K:2n/(10 + 2n) : [Serf, [0, 1, 945, -1333]]
More familial double-factorial sequences for the Mobius coefficients above.
K:(2 + 2n)/(-14 + 2n) : ['Serf', [-2, 251184, -16, -17553]]
K:(2 + 2n)/(-12 + 2n) : [Serf, [2, 19086, 14, -1533]]
K:(2 + 2n)/(-10 + 2n) : [Serf, [2, -1704, 12, 171]]
K:(2 + 2n)/(-8 + 2n) : [Serf, [2, 186, 10, -15]]
K:(2 + 2n)/(-6 + 2n) : [Serf, [2, -24, 8, 9]]
K:(2 + 2n)/(-4 + 2n) : [Serf, [2, 6, 6, 3]]
K:(2 + 2n)/(-2 + 2n) : [Serf, [2, 0, 4, 3]]
K:(2 + 2n)/(0 + 2n) : [Serf, [2, 2, 2, 1]]
K:(2 + 2n)/(2 + 2n) : [Serf, [2, 0, 0, 1]]
K:(2 + 2n)/(4 + 2n) : [Serf, [2, -2, -2, 3]]
K:(2 + 2n)/(6 + 2n) : [Serf, [6, -8, -12, 17]]
This one I don't fully get yet. Going up the second column from the split point (K:(2 + 2n)/(0 + 2n)), we have [2, 0, 6, -24, 186, ...], which is twice A263801, a sequence we've seen before (Partial sums of odd double factorials with alternating signs). The third column gong up is easy: it's just the even numbers. The fourth column going up from the split point is something new: [1, 3, 3, 9, -15, 171, -1533, 17553, ...]. Other than the {1} on the split point, they're all multiples of 3, but dividing by 3 gives us [1/3, 1, 1, 3, -5, 57, -511, -5851], which I can't figure out and OEIS hasn't heard of. It's growing about as fast as column 2, so maybe it's generated in a similar way. If it's also a sum of double factorial terms, then you'd think that subtracting the previous term from the current one would give you a new sequence that was a little easier to grok, but I'm still not grokking. Maybe Seiichi Manyama knows. I can't help but wonder if the function generating the Mobius coefficients is something I should be able to read off from the form of the polynomials. Like, oh, it's of order n over n? Try a factorial. 2n over 2n? Try a double factorial. 2n / n? Et cetera. Some of these PCFs are generating the same infinite tails and they only differ in the first few terms, and those terms are determining the Mobius transformation, yeah? I don't know. I'd like to know. I'm hopeful that I'll figure it out in time, but probably not soon.
You might think that extending the sequence down would be easy. With small numbers like [6, -8, 12, 17], how big could the next ones be? I'm here to tell you that at least one of them is big. If you want to find it, I'd appreciate it. It eludes me. I had a program running over night searching Mobius coefficients in the range (-600, 600), and it still hasn't found a solution. I think I'm going to cut it short and go to work, and then tonight I will figure out the coefficients myself, by hand, using my big brain. I can do this. I can do anything.
More families:
K:(4 + 2n)/(-8 + 2n) : [Serf, [65, 5, 29, -18]]
K:(4 + 2n)/(-6 + 2n) : [Serf, [7, -13, 68, 63]]
K:(4 + 2n)/(-4 + 2n) : [Serf, [16, 18, 25, 15]]
K:(4 + 2n)/(-2 + 2n) : [Serf, [12, 6, 13, 9]]
K:(4 + 2n)/(0 + 2n) : [Serf, [8, 6, 5, 3]]
K:(4 + 2n)/(2 + 2n) : [Serf, [4, 2, 1, 1]]
K:(4 + 2n)/(4 + 2n) : [Serf, [0, 2, 1, -1]]
K:(4 + 2n)/(6 + 2n) : [Serf, [-4, 6, 5, -7]]
K:(4 + 2n)/(8 + 2n) : [Serf, [24, -34, -39, 55]]
K:(6 + 2n)/(-4 + 2n) : [Serf, [-1, 89, 47, 21]]
K:(6 + 2n)/(-2 + 2n) : [Serf, [-13, 99, 21, 35]]
K:(6 + 2n)/(0 + 2n) : [Serf, [39, 27, 19, 12]]
K:(6 + 2n)/(2 + 2n) : [Serf, [15, 9, 4, 3]]
K:(6 + 2n)/(4 + 2n) : [Serf, [3, 3, 1, 0]]
K:(6 + 2n)/(6 + 2n) : [Serf, [3, -3, -2, 3]]
K:(6 + 2n)/(8 + 2n) : [Serf, [15, -21, -17, 24]]
K:(6 + 2n)/(10 + 2n) : [Serf, [39, -55, -56, 79]]
I'll try extending those from above and below. These are weird sequences so far though.
The remaining PCFs I've found that converge to transformations of Serf don't really fall into nice families yet:
K:(6 + -2n)/(0 + 4n) : [Serf, [0, 16, 0, 17]]
K:(6 + -2n)/(2 + -4n) : [Serf, [-2, 4, -7, 14]]
K:(8 + 2n)/(6 + 2n) : [Serf, [8, 0, -1, 3]]
K:(8 + 2n)/(8 + 2n) : [Serf, [-16, 24, 15, -21]]
K:(8 + 4n)/(-2 + 2n) : [Serf, [0, 12, 0, 7]]
K:(8 + 4n)/(-4 + 2n) : [Serf, [0, 16, 0, 13]]
K:-6/(8 + -4n) : [Serf, [9, 3, 18, 5]]
But maybe I just need to find their family members.
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I'm a little bored of the erf()s. I think I'm going to look for some {pi}s now. Let's look at (2nd/1st) order and (1st/2nd) order PCFs first. The Ramanujan Project's website has some documents with continued fractions that they claim converge to Mobius transformations of pi, but they don't give the polynomials for the fractions explicitly, and also if you compare documents, they repeat fractions across them but give different inequivalent expressions for the convergent values, which is a really bad sign. Like in one document they claim that a certain fraction converges to {pi/-2 + 1} and in another they say that same fraction converges to {pi/2 - 3}. Honestly, I wouldn't be surprised if the fraction were good, and the two sides of the equation were generated by different pieces of code and there's like an off-by-one error of whether they're evaluating the polynomials at 0 or something in one oft he pieces of code, but it's still kind of sad that they can't verify their equalities. So I think I'm going to ignore all of their shit and just try figure it out on my own.
Holy crow, if I ask Wolfram Alpha for continued fractions of constants, I get them, and lots of them are PCFs, and unlike the Ramanujan Project, I trust the results. I wouldn't have made this post if I'd have known about that. Did I waste a month of my life? Lol, no, it was more than a month. Also, not wasted; I've learned a lot and I still think there's a chance that some of the stuff in this post was not known before.
If I'm reading Wolfram Alpha correctly, I think K:(k/3 + k^2)/(4/3) might converge to a transformation of the cube root of 2. So that would be a polynomial with rational coefficients giving a cube root of two, whereas before I'd found one with transcendental coefficients. I saw a claim in a paper that cube roots can be done with PCFs in integer coefficients, but the paper had some mistakes and I'm not sure I trust any of it enough to bother working through their PCF finding procedure. It feels weird to dismiss the work of professional mathematicians who know so much more than me about these subjects than me, but, I mean, just like with the Ramanujan Project mathematicians, if your equalities aren't equal, or even close to equal, then your work isn't worth much, and it's worth even less to amateurs who can't figure out how to fix it.
Anyway, first let's do (2nd/1st) PCFs that converge to rational transformations of small rational powers of pi. I found 51 PCFs. I was feeling pretty good. But almost all were false positives. I'm not feeling so good anymore. I guess that's what I get for criticizing other people's equations. These seem real though:
K:(-1 + -3n + -2n^2)/(4 + 3n) = 2.751938393884109 : ['pi', [1, 0, 1, -2]]
K:(1 + 2n + n^2)/(3 + 2n) = 3.6597923663254877 : ['pi', [1, 0, -1, 4]]
K:(2n + n^2)/(3 + 2n) = 3.5038767877682173 : ['pi', [0, 4, 1, -2]]
K:(3n + -2n^2)/(1 + -3n) = 0.2800495767557787 : ['pi', [0, 2, 1, 4]]
K:(3n + -2n^2)/3n = 0.38898452964834274 : ['pi', [0, 2, 1, 2]]
And we've even got some families. Here's the first of four:
K:(-n + -2n^2)/(3 + 3n) = 2.329896183162744 : ['pi', [0, 2, -1, 4]]
K:(-n + -2n^2)/(4 + 3n) = 3.4769182395637523 : ['pi', [0, 2, -3, 10]]
K:(-n + -2n^2)/(5 + 3n) = 4.565635712908647 : ['pi', [0, 4, -15, 48]]
K:(-n + -2n^2)/(6 + 3n) = 5.626482121577372 : ['pi', [0, 12, -105, 332]]
K:(-n + -2n^2)/(7 + 3n) = 6.671352960735662 : ['pi', [0, 16, -315, 992]]
The first column of Mobius coefficients has 0s. I thought the second column would be twice {n!} = [1, 1, 2, 6, 24, 120, 720, 5040, 40320, ...] until that 16 showed up, and now I don't know what to think. I thought the third column would be negative A001147 = [1, 1, 3, 15, 105, 945, 10395, ...] (double factorial of odd numbers) until that -315 showed up, and now I don't know what to think. I thought the fourth column was twice A303108 = [2, 5, 24, 166, 1488, 16344, 212352, 3184560, ...] until the 992 showed up, and now I don't know what to think.
K:(n + -2n^2)/(1 + 3n) = 0.6366197723675814 : ['pi', [0, -2, -1, 0]]
K:(n + -2n^2)/(2 + 3n) = 1.7519383938841087 : ['pi', [0, -2, -1, 2]]
K:(n + -2n^2)/(3 + 3n) = 2.8074549930853796 : ['pi', [0, -4, -3, 8]]
K:(n + -2n^2)/(4 + 3n) = 3.84136469385785 : ['pi', [0, -12, -15, 44]]
K:(n + -2n^2)/(5 + 3n) = 4.864587800165587 : ['pi', [0, -48, -105, 320]]
K:(n + -2n^2)/(6 + 3n) = 5.8816238443659525 : ['pi', [0, -80, -315, 976]]
I had similar ideas about this family, but until my ideas let me find the next member or two of the sequence, I don't think they're worth mentioning. Second column might be twice A143383 = [1, 1, 2, 6, 24, 40, 240, 560, 13440, 120960, ...]. Third column here is the same as the third column in the last family. The fourth column appears to be the same as the fourth column of the last family minus the second column of the last family. I wonder if the coefficients would be simpler in terms of arcsin(1) = pi/2. For example,
K:(n + -2n^2)/(1 + 3n) = 0.6366197723675814 : ['asin(1)', [0, 1, 1, 0]]
K:(n + -2n^2)/(2 + 3n) = 1.7519383938841087 : ['asin(1)', [0, 1, 1, -1]]
...
And from the previous family:
K:(-n + -2n^2)/(3 + 3n) = 2.329896183162744 : ['asin(1)', [0, 1, -1, 2]]
...
Another family:
K:(1 + -n + -2n^2)/(2 + 3n) = 1.3889845296483427 : ['pi', [1, 4, 1, 2]]
K:(1 + -n + -2n^2)/(3 + 3n) = 2.5707963267948966 : ['pi', [1, 2, 0, 2]]
K:(1 + -n + -2n^2)/(4 + 3n) = 3.6597923663254877 : ['pi', [1, 0, -1, 4]]
K:(1 + -n + -2n^2)/(5 + 3n) = 4.715377359345629 : ['pi', [3, -4, -6, 20]]
K:(1 + -n + -2n^2)/(6 + 3n) = 5.754180950544863 : ['pi', [15, -32, -45, 144]]
K:(1 + -n + -2n^2)/(7 + 3n) = 6.783102651971715 : ['pi', [105, -272, -420, 1328]]
The last column might be twice A000698 = [1, 1, 2, 10, 74, 706, 8162, 110410, ....]. The [1, 6, 45] in the third column might be A001879 = [1, 6, 45, 420, 4725, 62370, 945945, 16216200, ...]. The linked page even mentions continued fraction expansions of pi.
K:(3 + n + -2n^2)/(0 + 3n) = 0.8970387201316157 : ['pi', [3, 8, 3, 10]]
K:(3 + n + -2n^2)/(1 + 3n) = 1.5814121072825869 : ['pi', [3, 10, 2, 6]]
K:(3 + n + -2n^2)/(2 + 3n) = 2.4399008464884426 : ['pi', [3, 8, 1, 4]]
K:(3 + n + -2n^2)/(3 + 3n) = 3.356194490192345 : ['pi', [3, 4, 0, 4]]
K:(3 + n + -2n^2)/(4 + 3n) = 4.300133700977887 : ['pi', [9, 0, -3, 16]]
K:(3 + n + -2n^2)/(5 + 3n) = 5.259717790119815 : ['pi', [45, -48, -30, 112]]
K:(3 + n + -2n^2)/(6 + 3n) = 6.229099207822988 : ['pi', [105, -192, -105, 352]]
Ooh, and here's a fifth family:
K:(n^2)/(1 + 2n) = 1.2732395447351628 : ['pi', [0, 4, 1, 0]]
K:(4n^2)/(2 + 4n) = 2.5464790894703255 : ['pi', [0, 8, 1, 0]]
K:(9n^2)/(3 + 6n) = 3.819718634205488 : ['pi', [0, 12, 1, 0]]
K:(16n^2)/(4 + 8n) = 5.092958178940651 : ['pi', [0, 16, 1, 0]]
K:(25n^2)/(5 + 10n) = 6.366197723675813 : ['pi', [0, 20, 1, 0]]
It seems K:((pn)^2)/(p (1 + 2n)) = 4p / pi. Nice.
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Okay, so I'm not done figuring out the patterns in the columns for the {pi} families, and I never looked very thoroughly for {pi}s in the (1st/2nd) order PCFs, but I wanted to do something new today. So I took the same set of (2nd/1st) PCFs and I started looking for other constants. This showed up right away:
K:(2n^2)/(2 + n) = 2.4663034623764317 : ['ln(3/2)', [0, 1, 1, 0]]
K:(3n^2)/(3 + 2n) = 3.4760594967822067 : ['ln(4/3)', [0, 1, 1, 0]]
K:(4n^2)/(4 + 3n) = 4.48142011772455 : ['ln(5/4)', [0, 1, 1, 0]]
K:(5n^2)/(5 + 4n) = 5.484814947747077 : ['ln(6/5)', [0, 1, 1, 0]]
K:(6n^2)/(6 + 5n) = 6.487159194630882 : ['ln(7/6)', [0, 1, 1, 0]]
K:(7n^2)/(7 + 6n) = 7.4888756894186175 : ['ln(8/7)', [0, 1, 1, 0]]
Pretty cool, right? It seems K:(pn^2)/(p + (p-1)n) = 1 / ln(p+1/p).
And here are some more:
K:(2 + 4n + 2n^2)/(3 + n) = 4.289052433381815 : ['ln(3/2)', [2, 0, -2, 1]]
K:(2n + 2n^2)/(4 + n) = 4.621171936693333 : ['ln(3/2)', [0, 1, 3, -1]]
K:(-2n + -2n^2)/(4 + 3n) = 3.2588913532709296 : ['ln(2)', [0, 1, -1, 1]]
K:(-4n + -4n^2)/(8 + 5n) = 7.301733334336725 : ['ln(4/3)', [0, -1, 3, -1]]
K:(-2n^2)/(2 + 3n) = 1.4426950408889634 : ['ln(2)', [0, 1, 1, 0]]
K:(-2n^2)/(3 + 3n) = 2.58869944956209 : ['ln(2)', [0, 1, 2, -1]]
K:(-2n^2)/(4 + 3n) = 3.6685303477828963 : ['ln(2)', [0, 2, 8, -5]]
K:(-2n^2)/(5 + 3n) = 4.7204521975582905 : ['ln(2)', [0, 3, 24, -16]]
K:(-3n^2)/(3 + 4n) = 2.4663034623764317 : ['ln(3/2)', [0, 1, 1, 0]]
K:(-3n^2)/(5 + 4n) = 4.621171936693333 : ['ln(3/2)', [0, 1, 3, -1]]
K:(-3n^2)/(7 + 4n) = 6.703043054709765 : ['ln(3/2)', [0, 2, 18, -7]]
K:(-4n^2)/(3 + 6n) = 2.4853397382384474 : ['ln(5/1)', [0, 4, 1, 0]]
K:(-4n^2)/(4 + 8n) = 3.6409569065073497 : ['ln(6/2)', [0, 4, 1, 0]]
K:(-4n^2)/(5 + 10n) = 4.720890004575315 : ['ln(7/3)', [0, 4, 1, 0]]
K:(-4n^2)/(6 + 12n) = 5.7707801635558535 : ['ln(8/4)', [0, 4, 1, 0]]
K:(-4n^2)/(4 + 5n) = 3.4760594967822067 : ['ln(4/3)', [0, 1, 1, 0]]
K:(-4n^2)/(7 + 5n) = 6.634454628786865 : ['ln(4/3)', [0, 1, 4, -1]]
K:(-4n^2)/(10 + 5n) = 9.716930346873635 : ['ln(4/3)', [0, 2, 32, -9]]
K:(-4n^2)/(4 + 5n) = 3.4760594967822067 : ['ln(4/3)', [0, 1, 1, 0]]
K:(-5n^2)/(5 + 6n) = 4.48142011772455 : ['ln(5/4)', [0, 1, 1, 0]]
K:(-6n^2)/(6 + 7n) = 5.484814947747077 : ['ln(6/5)', [0, 1, 1, 0]]
K:(-7n^2)/(7 + 8n) = 6.487159194630882 : ['ln(7/6)', [0, 1, 1, 0]]
K:(-8n^2)/(8 + 9n) = 7.4888756894186175 : ['ln(8/7)', [0, 1, 1, 0]]
K:(-6n^2)/(4 + 5n) = 3.08680830507151 : ['ln(3)', [0, 4, 3, -2]]
K:(-8n^2)/(4 + 6n) = 2.8853900817779268 : ['ln(2)', [0, 2, 1, 0]]
K:(-8n^2)/(6 + 6n) = 5.17739889912418 : ['ln(2)', [0, 2, 2, -1]]
K:(-8n^2)/(8 + 6n) = 7.337060695565793 : ['ln(2)', [0, 4, 8, -5]]
K:(-8n^2)/(10 + 6n) = 9.440904395116581 : ['ln(2)', [0, 3, 12, -8]]
K:(-n^2)/(2 + 4n) = 1.8204784532536749 : ['ln(3/1)', [0, 2, 1, 0]]
K:(-n^2)/(3 + 6n) = 2.8853900817779268 : ['ln(4/2)', [0, 2, 1, 0]]
K:(-n^2)/(4 + 8n) = 3.9152303779424353 : ['ln(5/3)', [0, 2, 1, 0]]
K:(-n^2)/(5 + 10n) = 4.932606924752863 : ['ln(6/4)', [0, 2, 1, 0]]
K:(-n^2)/(6 + 12n) = 5.944026823976923 : ['ln(7/5)', [0, 2, 1, 0]]
K:(2n^2)/(2 + n) = 2.4663034623764317 : ['ln(3/2)', [0, 1, 1, 0]]
K:(2n^2)/(5 + n) = 5.289052433381815 : ['ln(3/2)', [0, 1, -2, 1]]
K:(2n^2)/(8 + n) = 8.206109071150882 : ['ln(3/2)', [0, 2, 8, -3]]
K:(2n^2)/(11 + n) = 11.159161410188016 : ['ln(3/2)', [0, -3, 24, -10]]
K:(3n^2)/(3 + 2n) = 3.4760594967822067 : ['ln(4/3)', [0, 1, 1, 0]]
K:(3n^2)/(7 + 2n) = 7.301733334336725 : ['ln(4/3)', [0, -1, 3, -1]]
K:(3n^2)/(11 + 2n) = 11.21847791976103 : ['ln(4/3)', [0, 2, 18, -5]]
K:(4n^2)/(4 + 3n) = 4.48142011772455 : ['ln(5/4)', [0, 1, 1, 0]]
K:(5n^2)/(5 + 4n) = 5.484814947747077 : ['ln(6/5)', [0, 1, 1, 0]]
K:(6n^2)/(3 + n) = 3.9152303779424353 : ['ln(5/3)', [0, 2, 1, 0]]
K:(6n^2)/(8 + n) = 8.555726453801535 : ['ln(5/3)', [0, 4, -3, 2]]
K:(6n^2)/(13 + n) = 13.390676362996917 : ['ln(5/3)', [0, 8, 9, -4]]
K:(n^2)/5 = 5.177398899134019 : ['ln(2)', [0, 2, 2, -1]]
K:(n^2)/7 = 7.133372164200181 : ['ln(2)', [0, 6, -6, 5]]
K:(n^2)/9 = 9.106319696972268 : ['ln(2)', [0, 12, 12, -7]]
Lots of potential for generalization here.
Okay, so we just saw a lot of {pi}s and {log}s. What other constants do I still want to find expressions for? I still really want rational powers of integers, like 2^(1/3) and the Pisot numbers. I'd also like some values of trigonometric functions besides tan. like sin, cos, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, and atanh. I'd like to find Lambert's W(1), the Euler–Mascheroni γ and other Stieltjes constants, and the two famous continued fraction constants: Lévy's constant and Khinchin's constant. I was surprised that {Gamma()/Gamma()Gamma()} constants never showed up, but I don't really mind. I spent a while generating expressions from integrals, and it would be nice if some of those showed up, including some more expressions for the Euler-Gompertz constant. Finally, I want values of the Riemann zeta function and Dirichlet beta function, like Apery's constant and Catalan's constant. Some of those I know are doable. Some might not have expressions as PCFs with integer coefficients. We'll see.
K:(-1 + -2n + -n^2)/(4 + 2n) = 3.0947778327483753 : ['Euler-Gompertz constant', [1, 0, 2, -1]]
K:(-4n + -4n^2)/(6 + 4n) = 4.954755186317765 : ['Euler-Gompertz constant', [0, 2, -1, 1]]
K:(-4n^2)/(4 + 4n) = 3.3537500563574016 : ['Euler-Gompertz constant', [0, 2, 1, 0]]
K:(-4n^2)/(4 + 8n) = 3.6409569065073497 : ['atanh(1/2)', [0, 2, 1, 0]]
K:(-6n^2)/(4 + 5n) = 3.08680830507151 : ['atanh(1/2)', [0, 2, 3, -1]]
K:(-n + -n^2)/(3 + 2n) = 2.4773775931588826 : ['Euler-Gompertz constant', [0, 1, -1, 1]]
K:(-n^2)/(2 + 2n) = 1.6768750281787008 : ['Euler-Gompertz constant', [0, 1, 1, 0]]
K:(-n^2)/(2 + 4n) = 1.8204784532536749 : ['atanh(1/2)', [0, 1, 1, 0]]
K:(3n + 5n^2)/(6 + -n) = 0.48788029963573143 : ["Levy's constant", [2, -1, 5, -5]]
The constant {atan(1/2)} is just {ln(3)/2}, so it doesn't really count as a new constant, but I have a reason for using it. Do you see? Twice above, an expression for atan(1/2) shows up immediately after an expression for Euler-Gompertz, and with the same coefficients. That's interesting. What comes next in that sequence? I don't know, but I can calculate the values:
K:(-4n^2)/(4 + 4n) = 3.3537500563574016 : ['Euler-Gompertz constant', [0, 2, 1, 0]]
K:(-4n^2)/(4 + 8n) = 3.6409569065073497 : ['atanh(1/2)', [0, 2, 1, 0]]
K:(-4n^2)/(4 + 12n) = 3.740410611746318231477328577065701141012
so the next constant in the sequence will be 2 times the reciprocal of that {3.7404} number, or
? = 0.534700653911962488976762509564116040454
Let's compare that to the other pattern.
K:(-n^2)/(2 + 2n) = 1.6768750281787008 : ['Euler-Gompertz constant', [0, 1, 1, 0]]
K:(-n^2)/(2 + 4n) = 1.8204784532536749 : ['atanh(1/2)', [0, 1, 1, 0]]
K:(-n^2)/(2 + 6n) = 1.870205305873159115738664288532850570506
so the next constant in this sequence (hopefully the same sequence) will be the reciprocal of the {1.8702} number, or,
? = 0.534700653911962488976762509564116040454
You may notice that those numbers are perfectly equal to all decimal places. Kachow!
Let's look at the next number in the series!
From
K:(-4n^2)/(4 + 16n): 3.79536160560662022280395305889853814622
we use {2/3.79536...} to find
?_2 = 0.52695901150645065005114474277324342185.
Alternatively, from
K:(-n^2)/(2 + 8n) = 1.89768080280331011140197652944926907311
and {1/1.897...}, we find
?_2 = 0.52695901150645065005114474277324342185
which is again equal to all precision!
I have no idea what those two constants are! I would happily pay some small-to-moderate cash bounties for someone else to explain them to me.
I think now's a good time to clean up the earliest parts of this post. Like I never put the square roots or Bessel function PCFs into families.
For the square roots, i.e. the (0th/0th) order PCFS: When the numerator and denominator are non negative, we have one case for the even denominators:
K:p/2 = sqrt(p + 1) + 1
K:p/4 = sqrt(p + 4) + 2
K:p/6 = sqrt(p + 9) + 3
K:p/8 = sqrt(p + 16) + 4
Or more generally,
K:p/2r=sqrt(p+ r^2) + r
And one case for the odd denominators:
K:p/1 = (sqrt(4p + 1) + 1) / 2
K:p/3 = (sqrt(4p + 9) + 3) / 2
K:p/5 = (sqrt(4p + 25) + 5) / 2
K:p/7 = (sqrt(4p + 49) + 7) / 2
K:p/9 = (sqrt(4p + 81) + 9) / 2
Or more generally,
K:p/r = (sqrt(4p + r^2) + r) / 2
I don't know why there are two different cases, and I don't immediately see a way to unify them. I haven't looked at the patterns for negative numerators.
For the PCFs that converge to transformations of ratios of Bessel functions jv(,)/jv(,), the ratios are not at all independent. See for example
K:-1/(-2n) = 0.5750809150043059 : ['jv(0, 1)/jv(2, 1)', [1, 1, 2, 0]]
K:-1/(-2n) = 0.5750809150043059 : ['jv(1, 1)/jv(0, 1)', [1, 0, 0, 1]]
K:-1/(-2n) = 0.5750809150043059 : ['jv(2, 1)/jv(0, 1)', [1, 1, 0, 2]]
K:-1/(-2n) = 0.5750809150043059 : ['jv(3, 1)/jv(0, 1)', [1, 4, 0, 7]]
or
K:-1/-n = 2.575920321368222 : ['jv(1, 2)/jv(0, 2)', [1, 0, 0, 1]]
K:-1/-n = 2.575920321368222 : ['jv(2, 2)/jv(0, 2)', [1, 1, 0, 1]]
K:-1/-n = 2.575920321368222 : ['jv(3, 2)/jv(0, 2)', [1, 2, 0, 1]]
K:-1/-n = 2.575920321368222 : ['jv(4, 2)/jv(0, 2)', [1, 5, 0, 2]]
.
Rather than looking at the sequences in the columns of the Mobius coefficients for a single PCF, I'd like to start by finding ratios that make the Mobius coefficients [1, 0, 0, 1] for lots of PCFs.
K:-1/(0 + -2n) = 0.5750809150043059 : ['jv(1, 1)/jv(0, 1)', [1, 0, 0, 1]]
K:-1/(2 + -2n) = 0.26111426425529627 : ['jv(2, 1)/jv(1, 1)', [1, 0, 0, 1]]
K:-1/(4 + -2n) = 0.17025901341689473 : ['jv(3, 1)/jv(2, 1)', [1, 0, 0, 1]]
K:-1/(6 + -2n) = 0.12659582637537842 : ['jv(4, 1)/jv(3, 1)', [1, 0, 0, 1]]
K:-1/(8 + -2n) = 0.10084543360199086 : ['jv(5, 1)/jv(4, 1)', [1, 0, 0, 1]]
K:-1/(0 + -n) = 2.575920321368222 : ['jv(1, 2)/jv(0, 2)', [1, 0, 0, 1]]
K:-1/(1 + -n) = 0.6117892344322042 : ['jv(2, 2)/jv(1, 2)', [1, 0, 0, 1]]
K:-1/(2 + -n) = 0.3654501522438679 : ['jv(3, 2)/jv(2, 2)', [1, 0, 0, 1]]
K:-1/(3 + -n) = 0.2636486977499144 : ['jv(4, 2)/jv(3, 2)', [1, 0, 0, 1]]
K:-1/(4 + -n) = 0.20707400213083468 : ['jv(5, 2)/jv(4, 2)', [1, 0, 0, 1]]
K:-1/(5 + -n) = 0.17080855293377584 : ['jv(6, 2)/jv(5, 2)', [1, 0, 0, 1]]
K:-1/(6 + -n) = 0.14549223195099636 : ['jv(7, 2)/jv(6, 2)', [1, 0, 0, 1]]
K:-1/(7 + -n) = 0.1267808144093032 : ['jv(8, 2)/jv(7, 2)', [1, 0, 0, 1]]
K:-1/(8 + -n) = 0.11237122383858297 : ['jv(9, 2)/jv(8, 2)', [1, 0, 0, 1]]
K:-1/(1 + n) = 0.38821076556779577 : ['jv(0, 2)/jv(1, 2)', [1, 0, 0, 1]]
K:-1/(2 + n) = 1.634549847756132 : ['jv(1, 2)/jv(2, 2)', [1, 0, 0, 1]]
K:-1/(3 + n) = 2.736351302250086 : ['jv(2, 2)/jv(3, 2)', [1, 0, 0, 1]]
K:-1/(4 + n) = 3.7929259978691654 : ['jv(3, 2)/jv(4, 2)', [1, 0, 0, 1]]
K:-1/(5 + n) = 4.829191447066224 : ['jv(4, 2)/jv(5, 2)', [1, 0, 0, 1]]
K:-1/(6 + n) = 5.8545077680489985 : ['jv(5, 2)/jv(6, 2)', [1, 0, 0, 1]]
K:-1/(7 + n) = 6.873219185590445 : ['jv(6, 2)/jv(7, 2)', [1, 0, 0, 1]]
K:-1/(8 + n) = 7.887628776145773 : ['jv(7, 2)/jv(8, 2)', [1, 0, 0, 1]]
K:-1/(2 + 2n) = 1.7388857357447036 : ['jv(0, 1)/jv(1, 1)', [1, 0, 0, 1]]
K:-1/(4 + 2n) = 3.829740986583105 : ['jv(1, 1)/jv(2, 1)', [1, 0, 0, 1]]
K:-1/(6 + 2n) = 5.873404173624622 : ['jv(2, 1)/jv(3, 1)', [1, 0, 0, 1]]
K:-1/(8 + 2n) = 7.899154566398017 : ['jv(3, 1)/jv(4, 1)', [1, 0, 0, 1]]
Nice and orderly, right? And those families cover all the PCFs I've found that converge to transformations of jv(,)/jv(,), so I'm calling that success. I would't be at all surprised if there were other ones that didn't fall into those patterns, perhaps with a different constant in the numerator or with a different order, but I'm happy and satisfied with these patterns in their current state of development.
What about the PCFs that converged to ratios of modified Bessel functions, iv(,)/iv(.)? For (0th/1st), these patterns showed up right away:
K:1/(-4 + n) = 0.19369123409479713 : ['iv(5, 2)/iv(4, 2)', [1, 0, 0, 1]]
K:1/(-3 + n) = 0.23845341590004987 : ['iv(4, 2)/iv(3, 2)', [1, 0, 0, 1]]
K:1/(-2 + n) = 0.30878937306624005 : ['iv(3, 2)/iv(2, 2)', [1, 0, 0, 1]]
K:1/(-1 + n) = 0.43312742672231175 : ['iv(2, 2)/iv(1, 2)', [1, 0, 0, 1]]
K:1/(0 + n) = 0.697774657964008 : ['iv(1, 2)/iv(0, 2)', [1, 0, 0, 1]]
K:1/(1 + n) = 1.4331274267223117 : ['iv(0, 2)/iv(1, 2)', [1, 0, 0, 1]]
K:1/(2 + n) = 2.30878937306624 : ['iv(1, 2)/iv(2, 2)', [1, 0, 0, 1]]
K:1/(3 + n) = 3.2384534159000498 : ['iv(2, 2)/iv(3, 2)', [1, 0, 0, 1]]
K:1/(4 + n) = 4.193691234094797 : ['iv(3, 2)/iv(4, 2)', [1, 0, 0, 1]]
K:1/(-4 + 2n) = 0.16330611761053412 : ['iv(3, 1)/iv(2, 1)', [1, 0, 0, 1]]
K:1/(-2 + 2n) = 0.24019372387008975 : ['iv(2, 1)/iv(1, 1)', [1, 0, 0, 1]]
K:1/(0 + 2n) = 0.4463899658965345 : ['iv(1, 1)/iv(0, 1)', [1, 0, 0, 1]]
K:1/(2 + 2n) = 2.2401937238700897 : ['iv(0, 1)/iv(1, 1)', [1, 0, 0, 1]]
K:1/(4 + 2n) = 4.163306117610534 : ['iv(1, 1)/iv(2, 1)', [1, 0, 0, 1]]
But I'm still figuring out some others:
K:(-1 + -2n)/(-1 + -n) = 1.94837486112982 : ['iv(3, 1)/iv(2, 1)', [1, 6, 1, 3]]
K:(-1 + -2n)/(0 + -n) = 0.7336562064501496 : ['iv(1, 1)/iv(2, 1)', [1, 5, 3, 0]]
K:(-1 + -2n)/(2 + n) = 5.163306117610534 : ['iv(1, 1)/iv(2, 1)', [1, 1, 0, 1]]
K:(-1 + -2n)/(2 + n) = 5.163306117610534 : ['iv(2, 1)/iv(1, 1)', [1, 1, 1, 0]]
K:(-1 + -2n)/(2 + n) = 5.163306117610534 : ['iv(3, 1)/iv(2, 1)', [1, 5, 0, 1]]
K:(-1 + -2n)/(3 + n) = 1.806325641513043 : ['iv(0, 1)/iv(1, 1)', [1, 0, 1, -1]]
K:(-1 + -2n)/(4 + n) = 3.2401937238700897 : ['iv(1, 1)/iv(0, 1)', [1, 1, 1, 0]]
K:(-1 + -2n)/(4 + n) = 3.2401937238700897 : ['iv(2, 1)/iv(0, 1)', [-1, 3, -1, 1]]
K:(-1 + -2n)/(4 + n) = 3.2401937238700897 : ['iv(2, 1)/iv(1, 1)', [1, 3, 0, 1]]
K:(-1 + 2n)/(1 + -n) = 0.19367435848695685 : ['iv(0, 1)/iv(1, 1)', [-1, 2, -1, 1]]
K:(-1 + 2n)/(1 + -n) = 0.19367435848695685 : ['iv(2, 1)/iv(1, 1)', [1, 0, 1, 1]]
K:(-1 + 2n)/(1 + n) = 1.36303624396306 : ['iv(1, 1)/iv(2, 1)', [3, 0, 1, 5]]
K:(-1 + 2n)/(1 + n) = 1.3630362439630612 : ['iv(3, 1)/iv(0, 1)', [1, 4, -3, 3]]
K:(-1 + 2n)/(2 + -n) = 0.5536100341034655 : ['iv(0, 1)/iv(1, 1)', [1, -1, 1, 0]]
K:(-1 + 2n)/(2 + -n) = 0.5536100341034655 : ['iv(2, 1)/iv(0, 1)', [1, 1, 0, 2]]
K:(-1 + 2n)/(3 + -n) = 0.3086235223014997 : ['iv(1, 1)/iv(0, 1)', [1, 0, 1, 1]]
K:(-1 + 2n)/(4 + -n) = 0.22629672367078343 : ['iv(1, 1)/iv(2, 1)', [1, 1, 5, 2]]
K:(-1 + 2n)/(5 + -n) = 0.18103920383583277 : ['iv(0, 1)/iv(1, 1)', [-1, 4, 3, 3]]
K:(-2 + -4n)/(4 + n) = 2.893474032469793 : ['iv(4, 2)/iv(0, 2)', [1, 3, 2, 1]]
K:(-2 + -4n)/(6 + n) = 4.866254853444624 : ['iv(3, 2)/iv(1, 2)', [-1, 5, 0, 1]]
K:(-3 + -2n)/(2 + n) = 1.0175096930688106 : ['iv(0, 1)/iv(3, 1)', [1, 1, 1, 0]]
K:(-3 + -2n)/(2 + n) = 1.0175096930688106 : ['iv(3, 1)/iv(0, 1)', [1, 1, 0, 1]]
K:(-3 + -2n)/(4 + n) = 2.5132482562519423 : ['iv(1, 1)/iv(0, 1)', [3, -3, 3, -2]]
K:(-3 + 2n)/(3 + -n) = 1.193674358486957 : ['iv(0, 1)/iv(2, 1)', [1, 3, 1, 1]]
K:(-3 + 2n)/(4 + -n) = 0.7598062761299103 : ['iv(3, 1)/iv(1, 1)', [1, 3, 0, 4]]
K:(-3 + 2n)/(5 + -n) = 0.5810230754608696 : ['iv(3, 1)/iv(2, 1)', [0, 3, 1, 5]]
K:(1 + -2n)/(-1 + n) = 0.36303624396306 : ['iv(2, 1)/iv(1, 1)', [5, -2, -5, -1]]
K:(1 + -2n)/(-1 + n) = 0.3630362439630611 : ['iv(0, 1)/iv(3, 1)', [1, 4, 3, -3]]
K:(1 + -2n)/(0 + n) = 0.5132482562519421 : ['iv(1, 1)/iv(2, 1)', [1, -1, 1, 2]]
K:(1 + -2n)/(0 + n) = 0.5132482562519421 : ['iv(3, 1)/iv(0, 1)', [1, 1, -1, 2]]
K:(1 + -2n)/(1 + n) = 0.8063256415130431 : ['iv(1, 1)/iv(0, 1)', [1, 0, -1, 1]]
K:(1 + -2n)/(1 + n) = 0.8063256415130431 : ['iv(1, 1)/iv(2, 1)', [1, 0, 1, 1]]
K:(1 + -2n)/(2 + n) = 1.4463899658965345 : ['iv(0, 1)/iv(1, 1)', [1, 1, 1, 0]]
K:(1 + -2n)/(2 + n) = 1.4463899658965345 : ['iv(2, 1)/iv(1, 1)', [1, 3, 1, 2]]
K:(1 + -2n)/(3 + n) = 2.6913764776985003 : ['iv(1, 1)/iv(0, 1)', [2, 3, 1, 1]]
K:(1 + -2n)/(4 + n) = 3.7737032763292166 : ['iv(1, 1)/iv(0, 1)', [5, 7, 1, 2]]
K:(1 + 2n)/(-1 + n) = 0.6913764776985003 : ['iv(0, 1)/iv(1, 1)', [1, 0, 1, 1]]
K:(1 + 2n)/(-2 + n) = 0.37155708548628563 : ['iv(0, 1)/iv(1, 1)', [1, 1, 3, 2]]
K:(1 + 2n)/(-2 + n) = 0.37155708548628563 : ['iv(1, 1)/iv(0, 1)', [1, 1, 2, 3]]
K:(1 + 2n)/(-3 + n) = 0.2649916876805235 : ['iv(1, 1)/iv(0, 1)', [1, 2, 5, 7]]
K:(1 + 2n)/(0 + n) = 1.2401937238700897 : ['iv(1, 1)/iv(2, 1)', [1, 1, 1, 0]]
K:(1 + 2n)/(1 + n) = 1.9483748611298197 : ['iv(2, 1)/iv(1, 1)', [2, 1, -1, 1]]
K:(1 + 2n)/(1 + n) = 1.94837486112982 : ['iv(0, 1)/iv(2, 1)', [1, 3, 1, -3]]
K:(1 + 2n)/(2 + n) = 2.7545459072724148 : ['iv(2, 1)/iv(1, 1)', [5, 1, -5, 2]]
K:(1 + 2n)/(2 + n) = 2.754545907272415 : ['iv(1, 1)/iv(2, 1)', [1, 5, 2, -5]]
K:(2 + -4n)/(2 + n) = 1.547754433685334 : ['iv(3, 2)/iv(1, 2)', [1, 1, -2, 1]]
K:(2 + -4n)/(4 + n) = 3.395549315928016 : ['iv(2, 2)/iv(0, 2)', [-2, 4, 0, 1]]
K:(2 + 4n)/(-1 + n) = 1.292194650825733 : ['iv(3, 2)/iv(2, 2)', [-1, 2, 1, 1]]
K:(2 + 4n)/(-1 + n) = 1.292194650825733 : ['iv(4, 2)/iv(2, 2)', [1, 5, -1, 4]]
K:(2 + 4n)/(-3 + n) = 0.5890063179522377 : ['iv(0, 2)/iv(1, 2)', [1, 0, 1, 1]]
K:(2 + 4n)/(-3 + n) = 0.5890063179522377 : ['iv(3, 2)/iv(2, 2)', [1, 3, 2, 5]]
K:(3 + -2n)/(-2 + n) = 0.7545459072724151 : ['iv(3, 1)/iv(2, 1)', [-3, 3, 2, 3]]
K:(3 + -2n)/(0 + n) = 1.2401937238700897 : ['iv(1, 1)/iv(2, 1)', [1, 1, 1, 0]]
K:(3 + -2n)/(0 + n) = 1.2401937238700897 : ['iv(2, 1)/iv(0, 1)', [1, 1, -1, 1]]
K:(3 + -2n)/(0 + n) = 1.2401937238700897 : ['iv(3, 1)/iv(2, 1)', [1, 5, 1, 4]]
K:(3 + -2n)/(2 + n) = 2.3715570854862857 : ['iv(1, 1)/iv(0, 1)', [5, 7, 2, 3]]
K:(3 + 2n)/(0 + n) = 1.7737032763292164 : ['iv(1, 1)/iv(0, 1)', [3, 3, 1, 2]]
K:(3 + 2n)/(2 + n) = 3.163306117610534 : ['iv(3, 1)/iv(1, 1)', [1, 3, -1, 1]]
K:(3 + 2n)/(4 + n) = 4.835278195490148 : ['iv(2, 1)/iv(3, 1)', [3, 7, 2, -7]]
K:4/(-1 + n) = 1.3160945347187192 : ['iv(2, 4)/iv(0, 4)', [1, 0, -1, 1]]
K:4/(-1 + n) = 1.3160945347187192 : ['iv(3, 4)/iv(0, 4)', [-1, 2, 1, 1]]
K:4/(-1 + n) = 1.3160945347187192 : ['iv(5, 4)/iv(1, 4)', [-1, 4, 0, 3]]
K:4/(-2 + n) = 1.0392953503563447 : ['iv(5, 4)/iv(2, 4)', [1, 2, 0, 2]]
K:4/(-3 + n) = 0.8487615658325752 : ['iv(1, 4)/iv(4, 4)', [0, 5, 1, -1]]
K:4/(1 + n) = 2.316094534718719 : ['iv(0, 4)/iv(2, 4)', [1, 0, 1, -1]]
K:4/(1 + n) = 2.316094534718719 : ['iv(3, 4)/iv(2, 4)', [1, 3, 1, 1]]
K:4/(2 + n) = 3.0392953503563445 : ['iv(2, 4)/iv(1, 4)', [0, 2, 1, 0]]
K:4/(3 + n) = 3.848761565832575 : ['iv(5, 4)/iv(3, 4)', [-1, 4, 0, 1]]
I've been working on this project all month, and with the month's end, I think I'm done too. See you next time, folks.
Or maybe I'll just leave little results here occasionally.
K:(n^2)/(2 + 4n) = 2.15681043229161 : ['atan(1/2)', [0, 1, 1, 0]]
K:(-n + 2n^2)/(3n + 2n^2) = 0.1847550854110689 : ['cos(sqrt(2))', [1, 0, -1, 1]]
Oh weird, I tried ~8k PCFs with cubics in the numerator or denominator and the only result I got was that cos(sqrt(2)) thing, which is probably a false positive.
Today I thought that it might be interesting to find families of related PCFs even when I don't know the underlying constant. Ooh, and then maybe there will be a version of the underlying constant that makes the columns of Mobius coefficient simple, and that will help me hone my search for an alternative representation!
For example,
K:2/(-2 + 2n) = 0.4629300057070971 : ['T2', [-2, 2, 1, 0]]
K:2/(0 + 2n) = 0.8120409412226914 : ['T2', [1, 0, 0, 1]]
K:2/(2 + 2n) = 2.462930005707097 : ['T2', [0, 2, 1, 0]]
K:2/(4 + 2n) = 4.320307552639893 : ['T2', [1, 0, -1, 1]]
K:2/(6 + 2n) = 6.24399888019035 : ['T2', [-2, 2, 5, -4]]
These look like a family, but representing the family in terms of T2 makes for some weird column, where things are bouncing back and forth instead of strictly increasing in magnitude. I could try to fix that!
And here are some more families:
K:2/(-4 + 2n) = 0.3203075526398924 : ['T2', [5, -4, -1, 1]]
K:2/(-3 + n) = 0.4570348441460831 : ['T3', [-7, 8, 2, -2]]
K:2/(-2 + n) = 0.5785304719698354 : ['T3', [-4, 4, 1, -2]]
K:2/(-1 + n) = 0.7756355884645123 : ['T3', [-1, 2, 1, 0]]
K:2/(0 + n) = 1.1263572396234227 : ['T3', [1, 0, 0, 1]]
K:2/(1 + n) = 1.7756355884645123 : ['T3', [0, 2, 1, 0]]
K:2/(2 + n) = 2.5785304719698354 : ['T3', [2, 0, -1, 2]]
K:2/(3 + n) = 3.457034844146083 : ['T3', [1, -2, -2, 2]]
K:2/(4 + n) = 4.37603396243621 : ['T3', [4, -4, -7, 8]]
K:4n/(0 + 4n) = 0.8169686589393717 : ['T1', [2, 0, 0, 1]]
K:4n/(4 + 4n) = 4.463545173221204 : ['T1', [2, 0, -2, 1]]
K:4n/(8 + 4n) = 8.320449272538141 : ['T1', [2, 0, -12, 5]]
K:4n/(12 + 4n) = 12.244044390350753 : ['T1', [2, 0, -110, 45]]
K:n/(0 + 2n) = 0.4084843294696858 : ['T1', [1, 0, 0, 1]]
K:n/(2 + 2n) = 2.231772586610602 : ['T1', [1, 0, -2, 1]]
K:n/(4 + 2n) = 4.160224636269071 : ['T1', [1, 0, -12, 5]]
K:n/(6 + 2n) = 6.122022195175377 : ['T1', [1, 0, -110, 45]]
K:n/(2 + -2n) = 0.6744491614535594 : ['T1', [3, 0, 2, 1]]
K:4n/(4 + -4n) = 1.3488983229071188 : ['T1', [6, 0, 2, 1]]
K:9n/(6 + -6n) = 2.023347484360678 : ['T1', [9, 0, 2, 1]]
K:(16n)/(8 + -8n) = 2.6977966458142375 : ['T1', [12, 0, 2, 1]]
K:(-1 + -4n)/(6 + 4n) = 5.463545173221204 : ['T1', [0, 1, -2, 1]]
K:(-1 + -4n)/(10 + 4n) = 9.629148206212028 : ['T1', [1, 0, 5, -2]]
K:(-1 + -4n)/(14 + 4n) = 13.715604878167504 : ['T1', [2, -1, 22, -9]]
K:(4 + 4n)/(0 + 4n) = 1.508806710267816 : ['T1', [4, 2, 1, 2]]
K:(4 + 4n)/(4 + 4n) = 4.89614865421275 : ['T1', [0, 2, 1, 0]]
K:(4 + 4n)/(8 + 4n) = 8.629148206212028 : ['T1', [-4, 2, 5, -2]]
K:(4 + 4n)/(12 + 4n) = 12.482474896315763 : ['T1', [-24, 10, 49, -20]]
K:(3 + -4n)/(2 + 4n) = 1.8169686589393717 : ['T1', [2, 1, 0, 1]]
K:(3 + -4n)/(6 + 4n) = 5.89614865421275 : ['T1', [1, 2, 1, 0]]
K:(3 + -4n)/(10 + 4n) = 9.927090346442409 : ['T1', [2, 1, -2, 1]]
K:(3 + -4n)/(14 + 4n) = 13.943722309318042 : ['T1', [-1, 1, 5, -2]]
K:(1 + n)/(2 + -2n) = 0.10491444571017837 : ['T1', [4, -1, 5, 4]]
K:(1 + n)/(2 + 2n) = 2.448074327106375 : ['T1', [0, 1, 1, 0]]
K:(1 + n)/(0 + 2n) = 0.754403355133908 : ['T1', [2, 1, 1, 2]]
K:(2 + n)/(2 + 2n) = 2.6511016770928815 : ['T1', [2, 4, 2, 1]]
K:(3 + n)/(4 + 2n) = 4.607575292072305 : ['T1', [6, 3, -2, 2]]
K:(4 + n)/(6 + 2n) = 6.583546191216707 : ['T1', [8, -8, -14, 5]]
K:(1 + n)/2 = 2.679416883955586 : ['T5', [0, 1, 1, -2]]
K:(2 + n)/2 = 2.943700763448707 : ['T5', [2, -4, -2, 5]]
K:(1 + n)/2 = 2.679416883955586 : ['T5', [0, 1, 1, -2]]
K:(4 + 4n)/4 = 5.358833767911172 : ['T5', [0, 2, 1, -2]]
K:n/2 = 2.373215532822841 : ['T5', [1, 0, 0, 1]]
K:4n/4 = 4.746431065645682 : ['T5', [2, 0, 0, 1]]
K:(1 + n)/(4 + 2n) = 4.314574103106014 : ['T1', [-2, 1, 5, -2]]
K:(2 + n)/(4 + 2n) = 4.463545173221204 : ['T1', [2, 0, -2, 1]]
K:(7 + -4n)/(2 + 4n) = 2.508806710267816 : ['T1', [5, 4, 1, 2]]
K:(7 + -4n)/(10 + 4n) = 10.21515058414461 : ['T1', [5, 4, -1, 1]]
And this one stands alone for now:
K:(7 + -4n)/(2 + -4n) = 0.3488983229071187 : ['T1', [4, -1, 2, 1]]
Ripe for expansion, transformation, and generalization! Also, where are all the PCFs that converge to expressions for T4? I didn't even get the original PCF (namely K:2n/1) that I used to define T4 as a result of my search. Something's screwy. I bet I can find some T4 families once I unscrew it. Oh, I figured it out. I'll do it tomorrow. Oh weird, I thought I fixed the problem, but it's still not right. But I found these now:
K:(-1 + -4n)/(10 + 4n) = 9.629148206212028 : ['T1', [1, 0, 5, -2]]
K:(-1 + -4n)/(6 + 4n) = 5.463545173221204 : ['T1', [0, -1, 2, -1]]
So that's a start toward something.
Oh, and I had the best idea for extending families! Sometimes searching gets hard when the mobius coefficients get large, right? Like if you search all four in a range from -500 to 500, then your search space is like 1001^4 transformations of your constants to compare to your PCF values. But if you use the pcf value of the last member of your PCF family as your constant, then maybe constants will be smaller! Instead of looking fort big transformations of pi, look for small transformations of [pi, [14, -300, 26, 502] or whatever! Peel back the layers! It might just work!
Oh, poo, I think I figured it out why I wasn't getting any T4 families. T4 converges pretty slowly. If I evaluate 200 terms on the numerator and denominator of PCF and then I evaluate 500 terms, the numbers only match to the first 7 decimal places. The value of T4 that I was using in my constants dictionary was pretty high precision, but the PCF values that I was trying to find expressions for were only evaluated up to 200 terms. I think most PCFs converge really fast and 200 terms is overkill, but it's not overkill not for this one, apparently.
100 terms: 1.8327102532380870491593018482731165871: "K:2n/1",
200 terms: 1.8327056551794740375216842835896834886: "K:2n/1",
500 terms: 1.8327056412988286943191237676970854102: "K:2n/1",
10k terms: 1.8327056412986984162739589723713126654: "K:2n/1",
20k terms: 1.8327056412986984162739589723713126654: "K:2n/1",
So .... to find T4 families, I think it will suffice to use a 200-term constant and run t against 200-term PCF values, but if that fails, I might try looking at higher precision PCF values.
Wow, crazy, still no T4 families. Maybe they just don't exist. Even if I turn up the Mobius coefficient range and turn down the required precision for a match, I only get:
K:2n/1 = 1.832705655179474 : ['T4', [1, 0, 0, 1]]
K:-1/(2 + 4n) = 1.830487721712452 : ['T4', [4, 7, 7, -5]]
K:n/(2 + -2n) = 0.6744491614535594 : ['T4', [2, 6, 4, 7]]
I have no explanation for why all the other constants came in families and this one doesn't.
Whoa! New pattern! Possibly way more general than a PCF family:
K:(-4n)/(-4n) = 1.4079930916443677 : ['K:-n/(-2n)', [2, 0, 0, 1]]
K:(-4n)/(-6n) = 0.7557910294853206 : ['K:-n/(-3n)', [2, 0, 0, 1]]
K:(-4n)/(-8n) = 0.5345084925657189 : ['K:-n/(-4n)', [2, 0, 0, 1]]
K:4n/4n = 0.8169686589393717 : ['K:n/2n', [2, 0, 0, 1]]
K:4n/6 = 6.566197309860873 : ['K:n/3', [2, 0, 0, 1]]
K:4n/6n = 0.6030692070438289 : ['K:n/3n', [2, 0, 0, 1]]
K:4n/8 = 8.451214288978942 : ['K:n/4', [2, 0, 0, 1]]
K:4n/8n = 0.47141215039913276 : ['K:n/4n', [2, 0, 0, 1]]
For these ones at least,
K:(4n*numerator(n)) /(2*denominator(n)) = 2 * (K:numerator(n)/denominator(n))
I've wanted a pattern like this for a long time. If it holds more generally, it lets me construct infinite families at will of arbitrary degrees.
For example, the constant T4 that I couldn't find any family members for?
T4 = K:2n/1 = 1.8327056412986...
Well, maybe I just need to look at
K:2n/1
K:8n^2/2
K:32n^3/4
K:128n^4/8
et cetera!
Unfortunately, it seems that
K:(8n^2)/2
K:(32n^3)/4
do not converge in any reasonable amount of time, if ever. Evaluating 5000 terms versus 6000 terms, they each agree with themselves to fewer than four decimal places. Still no families for T4.
What would make me really happy is if I had a way of saying "add one to the numerator of the PCF, and the value will change in this way. Multiply the numerator by n and the value will change in this other way." and likewise for denominators. Then I'd be able to say what the value of any PCF would be by starting from K:1/1 and applying those rules repeatedly to make the desired PCFs, transforming phi along the way into tangents and logs and erf()s and what not. But it doesn't work even with the pattern above. K:4n/2 != 2 * phi.
So when does the pattern above work?
In all those examples, the original numerator was n or -n, and it was only -n when the denominator was also negative for positive values of n. While the positive cases can have constants in the numerator, it looks like the negative cases can't. So next I'd be tempted to examine:
K:n/5
K:n/5n
K:n/6
K:n/6n
and
K:-n/(-5n)
K:-n/(-6n)
K:-n/(-7n)
.
Also I want to see if applying the rule repeatedly works for the cases already discovered, even though it didn't work for T4.
