Bill Gosper mentioned: "The 549 move forced mate http://timkr.home.xs4all.nl/chess2/diary.htm" Wow, this is truly remarkable. I wouldn't have imagined the possibility of chess endgames taking that long to reach a conclusion (assuming perfect play). Apparently all 7-piece endgames have now been catalogued. Eugene Salamin wrote: "The series sum(x^k/2^k, k=0..inf) = 1/(1 - (x/2)) satisfies Julian's requirement." That's a rational function/number, though, so it's not as interesting as an irrational example (since the binary expansion must be periodic). A slightly less trivial rational number is this binary counter: Image[Partition[RealDigits[1/(1 - ((1/2)^20))^2 - 1/2, 2, 10000][[1]], 20]] And a one-digit modification to display the triangular numbers in binary: Image[Partition[RealDigits[1/(1 - ((1/2)^20))^3 - 1/2, 2, 10000][[1]], 20]] Simon Plouffe wrote: "a very good question now : can this be generated with a cellular automata ?" No, it cannot be generated in the obvious way as the Wolfram-style output of a one-dimensional cellular automaton. If it could, then the bitstring would become eventually periodic (when the CA is restricted to a finite world, as is the case for your algebraic constant). But sqrt(sqrt(2^(2^n) + 1)) is irrational, so the bitstring is aperiodic. #Contradiction Sincerely, Adam P. Goucher

----- Original Message ----- From: Eugene Salamin Sent: 03/27/14 07:47 PM To: math-fun Subject: Re: [math-fun] a strange class of algebraic numbers

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________________________________ 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.

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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

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