A025600 Number of n-move knight paths on 8 X 8 board from given corner to same corner.

1, 0, 2, 0, 16, 0, 264, 0, 6828, 0, 218192, 0, 7555444, 0, 269039512, 0, 9671837852, 0, 348829877760, 0, 12595130308612, 0, 454944702478600, 0, 16435098767896556, 0, 593753325451468144, 0, 21450960845508768532, 0, 774978877336933136632, 0

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Programs

Maple

b:= proc(n, i, j) option remember;
      `if`(n<0 or i<0 or i>7 or j<0 or j>7, 0, `if`({n, i, j}={0},
      1, add(b(n-1, i+r[1], j+r[2]), r=[[1, 2], [1, -2], [-1, 2],
      [-1, -2], [2, 1], [2, -1], [-2, 1], [-2, -1]])))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 28 2012

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[n<0 || i<0 || i>7 || j<0 || j>7, 0, If[Union[{n, i, j}] == {0}, 1, Sum[b[n-1, i+r[[1]], j+r[[2]]], {r, {{1, 2}, {1, -2}, {-1, 2}, {-1, -2}, {2, 1}, {2, -1}, {-2, 1}, {-2, -1}}}]]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Formula

From Vaclav Kotesovec, Nov 26 2012: (Start)

G.f.: 1 - 2495/3704 - (-2495 + 257062*x^2 - 10940636*x^4 + 261002480*x^6 - 3944912606*x^8 + 40234628876*x^10 - 286888584304*x^12 + 1458140925208*x^14 - 5325997352347*x^16 + 13961752450926*x^18 - 25982840678332*x^20 + 33572692661080*x^22 - 28997305139008*x^24 + 15706751871616*x^26 - 4743107684352*x^28 + 598878986240*x^30)/(3704*(-1+x)*(1+x)*(-1+2*x)*(1+2*x)*(1 - 3*x - 27*x^2 + 29*x^3 + 162*x^4 - 42*x^5 - 276*x^6 - 16*x^7 + 96*x^8)*(1 + 3*x - 27*x^2 - 29*x^3 + 162*x^4 + 42*x^5 - 276*x^6 + 16*x^7 + 96*x^8)*(1 - 38*x^2 + 546*x^4 - 3712*x^6 + 12253*x^8 - 17754*x^10 + 7408*x^12))

Nonzero terms a(n+2)/a(n) tends to 36.12804064450295915...

(End)