A025604 Number of n-move queen paths on 8x8 board from given corner to same corner.

1, 0, 21, 180, 3699, 78186, 1763689, 40219584, 921030783, 21114454518, 484210273013, 11105346505100, 254708804007115, 5841980066693506, 133991588716915713, 3073232605116594392, 70487720404861582487, 1616707836356340649102, 37080846132267449237645

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..736

Formula

From Vaclav Kotesovec, Nov 26 2012: (Start)

G.f.: ((-1+3*x)*(-1 + 65*x - 1704*x^2 + 23568*x^3 - 178704*x^4 + 576430*x^5 + 1641524*x^6 - 22152084*x^7 + 63390443*x^8 + 107381625*x^9 - 1072548108*x^10 + 1604228092*x^11 + 5065000686*x^12 - 19304907936*x^13 + 3650528536*x^14 + 75404779968*x^15 - 99704483296*x^16 - 97719166208*x^17 + 292843382912*x^18 - 76390477824*x^19 - 291118184448*x^20 + 242903138304*x^21 + 32643219456*x^22 - 93923573760*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, 0, 0):
seq(a(n), n=0..20);  # 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[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, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A254681 A219625 A244875 * A219412 A062159 A059721

Adjacent sequences: A025601 A025602 A025603 * A025605 A025606 A025607

Keyword

nonn,easy

Author

David W. Wilson

Status

approved