A125681 Decimal expansion of a non-holonomic random walk constant.

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

Offset

0,2

Comments

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].

Mishna and Rechnitzer use nonstandard indices for Fibonacci numbers. - R. J. Mathar, Nov 04 2007

Links

Table of n, a(n) for n=0..90.

Marni Mishna and Andrew Rechnitzer, Two Non-holonomic Lattice Walks in the Quarter Plane, arXiv:math/0701800

Formula

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

Example

0.173178883...

Maple

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

Mathematica

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 *)

Crossrefs

Sequence in context: A178149 A318669 A110636 * A021899 A176435 A133722

Adjacent sequences: A125678 A125679 A125680 * A125682 A125683 A125684

Keyword

cons,nonn

Author

Jonathan Vos Post, Jan 30 2007

Extensions

Corrected and extended by R. J. Mathar, Nov 04 2007

Status

approved