________________________________ From: Julian Ziegler Hunts <julianj.zh@gmail.com> To: Simon Plouffe <simon.plouffe@gmail.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Thursday, March 27, 2014 11:40 AM Subject: Re: [math-fun] a strange class of algebraic numbers
The two functions are the same, after replacing k by 2s+1. Even k/half-integral s only produce a little bit of regularity, namely the first few bits because it's very close to 1, but odd k/integral s produce the longer (length quadratic in k) patterns.
do you see any way to lengthen the nonrandomness?
Find some other algebraic function whose Taylor series' coefficients have power-of-two denominators and numerators that grow more slowly (sub-exponentially, if possible)? The actual size of the denominators doesn't matter too much, unless they grow really quickly, since only the ratio of two consecutive denominators contributes (currently, the denominators grow exponentially, hence contribute only a constant (because the numerators are also exponential) amount to the non-randomness). I don't know how you would go about constructing such a function, though.
Julian --------------------------------------------- The series sum(x^k/2^k, k=0..inf) = 1/(1 - (x/2)) satisfies Julian's requirement.
-- Gene