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A125681
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Decimal expansion of a non-holonomic random walk constant.
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0
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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; internal format)
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OFFSET
| 0,2
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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 non-standard indices for Fibonacci numbers. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 04 2007
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LINKS
| Marni Mishna and Andrew Rechnitzer, Two Non-holonomic Lattice Walks in the Quarter Plane, arXiv:math/0701800
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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 (mathar(AT)strw.leidenuniv.nl), Nov 04 2007
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EXAMPLE
| 0.173178883...
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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 (mathar(AT)strw.leidenuniv.nl), Nov 04 2007
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CROSSREFS
| Sequence in context: A199722 A178149 A110636 * A021899 A176435 A133722
Adjacent sequences: A125678 A125679 A125680 * A125682 A125683 A125684
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KEYWORD
| cons,nonn
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AUTHOR
| Jonathan Vos Post (jvospost3(AT)gmail.com), Jan 30 2007
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EXTENSIONS
| Corrected and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 04 2007
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