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%I #23 Aug 09 2015 23:43:13
%S 0,0,0,0,0,0,108,0,5124,0,198364,0,7316404,0,266099052,0,9635242692,0,
%T 348371291676,0,12589359561300,0,454871893274732,0,16434178541015076,
%U 0,593741680769005980,0,21450813362014160244,0,774977008153111237100,0
%N Number of n-move knight paths on 8 X 8 board from given corner to opposite corner.
%H Alois P. Heinz, <a href="/A025601/b025601.txt">Table of n, a(n) for n = 0..400</a>
%F From _Vaclav Kotesovec_, Nov 26 2012: (Start)
%F 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))
%F Nonzero terms a(n+2)/a(n) tends to 36.12804064450295915...
%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(b(n-1, i+r[1], j+r[2]), r=[[1, 2], [1, -2], [-1, 2],
%p [-1, -2], [2, 1], [2, -1], [-2, 1], [-2, -1]])))
%p end:
%p a:= n-> b(n, 7, 7):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Jun 28 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[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_ *)
%K nonn
%O 0,7
%A _David W. Wilson_
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