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

0,3

Links

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

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)

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 *)

Crossrefs

Sequence in context: A111978 A146558 A364514 * A009006 A155585 A350972

Adjacent sequences: A025597 A025598 A025599 * A025601 A025602 A025603

Keyword

nonn

Author

David W. Wilson

Status

approved