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A094061
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Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard.
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2
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1, 0, 8, 24, 216, 1200, 8840, 58800, 423640, 3000480, 21824208, 158964960, 1171230984, 8668531872, 64574844048, 483114856224
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Analog of A054474 for walks on a square grid where the steps can be made diagonally as well.
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REFERENCES
| Joyner, D., "Adventures in Group Theory: Rubik's Cube, Merlin's Machine and Other Mathematical Toys", Johns Hopkins University Press, 2002, pp. 79
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FORMULA
| (n+1)^2*a(n+1) = n*(5*n+1)*a(n) + 2*(15*n^2+6*n-5)*a(n-1) - 8*(5*n^2-23*n+21)*a(n-2) - 64*(n-2)^2*a(n-3).
Generating function: G(x) = (2/(Pi*(1+4*x))) * EllipticK(4*sqrt(x*(1+x))/(1+4*x))
= 1/(1+4*x) * hypergeom([1/2,1/2], [1], 16*(x*(1+x))/(1+4*x)^2) - Sergey Perepechko, Jan 15 2011
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MAPLE
| a:=array(0..30):a[0]:=1:a[1]:=0:a[2]:=8:a[3]:=24:for n from 3 to 29 do a[n+1]:= (n*(5*n+1)*a[n]+2*(15*n^2+6*n-5)*a[n-1]-8*(5*n^2-23*n+21)*a[n-2]-64*(n-2)^2*a[n-3])/(n+1)^2: print(n+1, a[n+1]) od:
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MATHEMATICA
| a[n_]:=Module[{f=(x+x^-1+y+y^-1+x y+x^-1y+x^-1y^-1+x y^-1)^n, s}, s=Series[f, {x, 0, 0}, {y, 0, 0}]; SeriesCoefficient[s, {0, 0}]] - Armin Vollmer (Armin.Vollmer(AT)kabelleipzig.de), May 01 2006
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CROSSREFS
| Cf. A098070.
Sequence in context: A100042 A061027 A052656 * A182589 A002268 A050893
Adjacent sequences: A094058 A094059 A094060 * A094062 A094063 A094064
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KEYWORD
| nonn
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AUTHOR
| Matthijs Coster (matthijs(AT)coster.demon.nl), Apr 29 2004
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EXTENSIONS
| More terms from and entry improved by Sergey Perepechko (persn(AT)aport.ru), Sep 06 2004
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