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A125681 Decimal expansion of a non-holonomic random walk constant. 1
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 (list; constant; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified June 25 04:06 EDT 2019. Contains 324345 sequences. (Running on oeis4.)