A025602 Number of n-move knight paths on 8x8 board from given corner to adjacent corner.

0, 0, 0, 0, 0, 18, 0, 852, 0, 32974, 0, 1216736, 0, 44265306, 0, 1602980908, 0, 57958913510, 0, 2094514154552, 0, 75677759126258, 0, 2734176449182980, 0, 98781516222538110, 0, 3568795954184265360, 0, 128933769847371450506, 0, 4658126510414596025052, 0

Offset

0,6

Links

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

Formula

From Vaclav Kotesovec, Nov 26 2012: (Start)

G.f.: -(2*x^5*(-9 + 186*x^2 - 1433*x^4 + 5172*x^6 - 9228*x^8 + 8048*x^10 - 3360*x^12 + 768*x^14))/((-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))

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, 7):
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, 7]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A376444 A376442 A221394 * A088126 A040327 A040328

Adjacent sequences: A025599 A025600 A025601 * A025603 A025604 A025605

Keyword

nonn

Author

David W. Wilson

Status

approved