A025605 Number of n-move queen paths on 8x8 board from given corner to opposite corner.

0, 1, 8, 157, 3362, 76367, 1748900, 40115777, 920244134, 21108602611, 484166076904, 11105013004773, 254706278398058, 5841960927064087, 133991443518199596, 3073231503116849801, 70487712038010275886, 1616707772818851078203, 37080845649697782625136

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..225

Formula

From Vaclav Kotesovec, Nov 26 2012: (Start)

G.f.: -(x*(-1 + 60*x - 1491*x^2 + 19722*x^3 - 141652*x^4 + 390120*x^5 + 1878634*x^6 - 19619492*x^7 + 49699215*x^8 + 120035252*x^9 - 974426255*x^10 + 1310451626*x^11 + 4951791366*x^12 - 17752007880*x^13 + 2466834328*x^14 + 71791048288*x^15 - 94759824992*x^16 - 93926071168*x^17 + 285741722496*x^18 - 78159888384*x^19 - 288554182656*x^20 + 244774010880*x^21 + 35092168704*x^22 - 97271414784*x^23 + 24970788864*x^24))/((1 - 8*x - 14*x^2 + 152*x^3 + 49*x^4 - 816*x^5 - 36*x^6 + 1152*x^7)*(-1 + 40*x - 482*x^2 + 2120*x^3 + 79*x^4 - 26128*x^5 + 55636*x^6 + 29184*x^7 - 170880*x^8 + 112896*x^9)*(-1 + 20*x - 130*x^2 + 168*x^3 + 1375*x^4 - 4652*x^5 - 348*x^6 + 15472*x^7 - 10816*x^8 - 7296*x^9 + 4608*x^10))

a(n+1)/a(n) tends to 22.936022136221...

(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 (add (b(n-1, i+t*r[1], j+t*r[2]), r=[[1, 1],
      [1, -1], [-1, 1], [-1, -1], [0, 1], [0, -1], [1, 0],
      [-1, 0]]), t=1..7)))
    end:
a:= n-> b (n, 7, 7):
seq (a(n), n=0..20);  # Alois P. Heinz, Jun 26 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[Sum[b[n-1, i+t*r[[1]], j+t*r[[2]]], {r, {{1, 1}, {1, -1}, {-1, 1}, {-1, -1}, {0, 1}, {0, -1}, {1, 0}, {-1, 0}}}], {t, 1, 7}]]]; 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: A221098 A171211 A211043 * A114223 A278669 A201030

Adjacent sequences: A025602 A025603 A025604 * A025606 A025607 A025608

Keyword

nonn

Author

David W. Wilson

Status

approved