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 On Thu, Mar 27, 2014 at 2:43 AM, Simon Plouffe <simon.plouffe@gmail.com>wrote:
Hello, the function you mention is not the same,
1+Sqrt[1+(Sqrt[1+4^(1-k)]-1)/2]/2^(k/2+1/2)
the original one posted is : 1 + 1/4*(2*4^s + 2*(16^s + 1)^(1/2))^(1/2)/(2^s)^2
which pushes the pattern very far when (here s) is equal to 100,1000 or 1000000, far : many billion bits. The one you mention only pushes up to a few hundreds bits I could see.
What type of gen. function would push the coefficients up to the billlions without being too big ? I don't see.
Best regards,
Simon Plouffe