A125681 - OEIS

%I #12 Aug 12 2015 21:06:06

%S 1,7,3,1,7,8,8,8,3,5,5,1,2,2,0,6,3,0,4,6,6,2,6,3,4,4,9,0,5,7,2,6,5,9,

%T 7,6,8,4,3,3,5,4,7,0,2,2,6,3,7,2,8,7,4,9,0,8,9,1,5,7,4,5,4,9,0,0,3,4,

%U 7,1,7,0,1,2,6,8,0,5,0,2,8,2,3,3,7,7,5,7,2,6,9

%N Decimal expansion of a non-holonomic random walk constant.

%C The number of walks of length n with step set {NE,SE,NW} confined to the quarter plane is asymptotic to Alpha*(3^n) + O(8^(n/2)), where Alpha is a constant given by 1-2SUM[n>=0]((-1)^n)/F(2n)F(2n+2) ~ 0.1731788836... [Corollary 2.7 of Mishna reference, p. 9].

%C Mishna and Rechnitzer use nonstandard indices for Fibonacci numbers. - _R. J. Mathar_, Nov 04 2007

%H Marni Mishna and Andrew Rechnitzer, <a href="http://arXiv.org/abs/math/0701800">Two Non-holonomic Lattice Walks in the Quarter Plane</a>, arXiv:math/0701800

%F 1-2*SUM[n>=0]((-1)^n)/F(2n)F(2n+2) where F(n) is the Fibonacci sequence. 1-2*SUM[n>=0]((-1)^n)/A000045(2n+1)*A000045(2n+3). - _R. J. Mathar_, Nov 04 2007

%e 0.173178883...

%p Digits := 100 : F := proc(n) combinat[fibonacci](n+1) ; end: s := 0 : for n from 0 do s := s+(-1)^n/F(2*n)/F(2*n+2) ; print(1.-2.*s) ; od: # _R. J. Mathar_, Nov 04 2007

%t digits = 100; rd[k_] := rd[k] = RealDigits[1 - 2*NSum[(-1)^n/(Fibonacci[2*n + 1]*Fibonacci[2*n + 3]), {n, 0, 2^k}, WorkingPrecision -> digits + 10]][[1]]; rd[k = 4]; While[rd[k] != rd[k - 1], k++]; A125681 = rd[k] (* _Jean-François Alcover_, Oct 30 2012 *)

%K cons,nonn

%O 0,2

%A _Jonathan Vos Post_, Jan 30 2007

%E Corrected and extended by _R. J. Mathar_, Nov 04 2007