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Search: id:A054474
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| A054474 |
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Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages. |
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+0
8
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| 1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432
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OFFSET
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0,2
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COMMENT
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1-dimensional and 3-dimensional analogues are A002420 and A049037.
Trajectories returning to the origin are prohibited, contrary to the situation in A094061.
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
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LINKS
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S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
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FORMULA
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G.f.: 2-AGM(1, (1-16x)^(1/2)).
Let (in Maple notation) G(x):=4/Pi*EllipticK(4*t)-2/Pi*EllipticF(1/sqrt(2+4*t), 4*t)-2/Pi*EllipticF(1/sqrt(2-4*t), 4*t), then GenFunc(x):=2-1/G(x) - Sergey Perepechko (persn(AT)aport.ru), Sep 11 2004
G.f.: 2-Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 23 2005
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EXAMPLE
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a(5)=22064 i.e. there are 22064 different walks (on a square lattice) that start and end at origin after 2*5=10 steps, avoiding origin at intermediate steps.
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PROGRAM
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(PARI) a(n)=if(n<0, 0, polcoeff(2-agm(1, sqrt(1-16*x+x*O(x^n))), n))
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CROSSREFS
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Cf. A002894, A002420, A049037.
Sequence in context: A065526 A032333 A068965 this_sequence A066917 A032081 A034235
Adjacent sequences: A054471 A054472 A054473 this_sequence A054475 A054476 A054477
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KEYWORD
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easy,nonn
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AUTHOR
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Alessandro Zinani (alzinani(AT)tin.it), May 19 2000
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