A025601 Number of n-move knight paths on 8 X 8 board from given corner to opposite corner.

0, 0, 0, 0, 0, 0, 108, 0, 5124, 0, 198364, 0, 7316404, 0, 266099052, 0, 9635242692, 0, 348371291676, 0, 12589359561300, 0, 454871893274732, 0, 16434178541015076, 0, 593741680769005980, 0, 21450813362014160244, 0, 774977008153111237100, 0

Offset

0,7

Links

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

Formula

From Vaclav Kotesovec, Nov 26 2012: (Start)

G.f.: -1209/3704 - (-1209 + 128154*x^2 - 5653284*x^4 + 140288880*x^6 - 2209572306*x^8 + 23499491668*x^10 - 174711214800*x^12 + 925284326152*x^14 - 3517813018109*x^16 + 9584107200498*x^18 - 18500002582980*x^20 + 24730494598568*x^22 - 22029459164480*x^24 + 12263416585088*x^26 - 3794033661952*x^28 + 489909739520*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, 7, 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, 7, 7]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A113471 A057388 A192071 * A109005 A036196 A304565

Adjacent sequences: A025598 A025599 A025600 * A025602 A025603 A025604

Keyword

nonn

Author

David W. Wilson

Status

approved