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%I #18 May 28 2015 04:53:19
%S 0,1,8,157,3362,76367,1748900,40115777,920244134,21108602611,
%T 484166076904,11105013004773,254706278398058,5841960927064087,
%U 133991443518199596,3073231503116849801,70487712038010275886,1616707772818851078203,37080845649697782625136
%N Number of n-move queen paths on 8x8 board from given corner to opposite corner.
%H Alois P. Heinz, <a href="/A025605/b025605.txt">Table of n, a(n) for n = 0..225</a>
%F From _Vaclav Kotesovec_, Nov 26 2012: (Start)
%F 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))
%F a(n+1)/a(n) tends to 22.936022136221...
%F (End)
%p b:= proc(n, i, j) option remember;
%p `if`(n<0 or i<0 or i>7 or j<0 or j>7, 0, `if`({n, i, j}={0},
%p 1, add (add (b(n-1, i+t*r[1], j+t*r[2]), r=[[1, 1],
%p [1, -1], [-1, 1], [-1, -1], [0, 1], [0, -1], [1, 0],
%p [-1, 0]]), t=1..7)))
%p end:
%p a:= n-> b (n, 7, 7):
%p seq (a(n), n=0..20); # _Alois P. Heinz_, Jun 26 2012
%t 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_ *)
%K nonn
%O 0,3
%A _David W. Wilson_
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