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A025601 Number of n-move knight paths on 8 X 8 board from given corner to opposite corner. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 27 21:14 EDT 2022. Contains 354110 sequences. (Running on oeis4.)