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A094061 Number of n-moves paths of a king starting and ending at the origin of an infinite chessboard. 8
1, 0, 8, 24, 216, 1200, 8840, 58800, 423640, 3000480, 21824208, 158964960, 1171230984, 8668531872, 64574844048, 483114856224, 3630440899800, 27379154692032, 207172490054816, 1572194644061184, 11962847247681616, 91242602561647680, 697438669619791008 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Analog of A054474 for walks on a square grid where the steps can be made diagonally as well.

REFERENCES

Joyner, D., "Adventures in Group Theory: Rubik's Cube, Merlin's Machine and Other Mathematical Toys", Johns Hopkins University Press, 2002, pp. 79

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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).

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) ~ 2^(3*n+1)/(3*Pi*n). - Vaclav Kotesovec, Aug 16 2013

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

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:

# 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:

seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

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

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 *)

PROG

(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))$

A094061(n):=a[n]$

makelist(A094061(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

CROSSREFS

Cf. A098070, A253974, A254129, A254459.

Sequence in context: A226962 A221784 A052656 * A182589 A002268 A050893

Adjacent sequences:  A094058 A094059 A094060 * A094062 A094063 A094064

KEYWORD

nonn,easy

AUTHOR

Matthijs Coster, Apr 29 2004

EXTENSIONS

More terms from and entry improved by Sergey Perepechko, Sep 06 2004

STATUS

approved

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Last modified February 20 18:39 EST 2019. Contains 320345 sequences. (Running on oeis4.)