OFFSET
0,3
COMMENTS
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
REFERENCES
D. Joyner, "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
Peter Bala, A note on A094061
FORMULA
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) ~ 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
a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A002426(k)^2. - Seiichi Manyama, Oct 29 2019
From Peter Bala, Feb 08 2022: (Start)
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
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:
# Alternative:
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}]]; Table[a[n], {n, 1, 22}] (* 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 */
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Matthijs Coster, Apr 29 2004
EXTENSIONS
More terms from and entry improved by Sergey Perepechko, Sep 06 2004
STATUS
approved