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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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21
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1, 0, 8, 24, 216, 1200, 8840, 58800, 423640, 3000480, 21824208, 158964960, 1171230984, 8668531872, 64574844048, 483114856224, 3630440899800, 27379154692032, 207172490054816, 1572194644061184, 11962847247681616, 91242602561647680, 697438669619791008
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OFFSET
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0,3
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COMMENTS
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The chessboard here is the full four-quadrant board Z X Z.
This is an analog of A054474 for walks on a square grid where the steps can be made diagonally as well.
a(n) is the constant term in the expansion of ((x + 1/x) * (y + 1/y) + x^2 + 1/x^2 + y^2 + 1/y^2)^n. - Seiichi Manyama, Nov 03 2019
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REFERENCES
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D. Joyner, "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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LINKS
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FORMULA
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D-finite with recurrence (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).
G.f.: (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
a(n) = (1/Pi^2) * Integral_{y = 0..Pi} Integral_{x = 0..Pi} (2*cos(x) + 2*cos(y) + 4*cos(x)*cos(y))^n dx dy. - Peter Bala, Feb 14 2017
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and positive integers n and k.
Conjecture: the stronger congruences a(n*p^k) == a(n*p^(k-1)) (mod p^(2*k)) hold for all primes p >= 5 and positive integers n and k. (End)
a(n) = Sum_{j = 0..n} Sum_{k = 0..j} binomial(2*j,j)^2*binomial(j,k)* binomial(n+j-k,2*j)*(-4)^(n-j-k). - Peter Bala, Mar 19 2022
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MAPLE
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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:
# second Maple program
a:= proc(n) option remember; `if`(n<3, (n-1)*(9*n-2)/2,
((n-1)*(3*n-1)*(3*n-4) *a(n-1)
+(108*n^3-396*n^2+452*n-152) *a(n-2)
+32*(3*n-2)*(n-2)^2 *a(n-3))/ (n^2*(3*n-5)))
end:
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MATHEMATICA
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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
CoefficientList[Series[1/(1+4*x)*LegendreP[-1/2, 1-32*x*(1+x)/(1+4*x)^2], {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 16 2013 *)
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PROG
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(Maxima)
a[0]:1$
a[1]:0$
a[2]:8$
a[3]:24$
a[n]:=((n-1)*(3*n-1)*(3*n-4) *a[n-1]
+(108*n^3-396*n^2+452*n-152) *a[n-2]
+32*(3*n-2)*(n-2)^2 *a[n-3])/ (n^2*(3*n-5))$
(PARI) {a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef((1+x+1/x)^k, 0)^2)} \\ Seiichi Manyama, Oct 29 2019
(PARI) {a(n) = polcoef(polcoef(((x+1/x)*(y+1/y)+x^2+1/x^2+y^2+1/y^2)^n, 0), 0)} \\ Seiichi Manyama, Nov 03 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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