%I #21 Aug 09 2015 23:42:57
%S 1,0,2,0,16,0,264,0,6828,0,218192,0,7555444,0,269039512,0,9671837852,
%T 0,348829877760,0,12595130308612,0,454944702478600,0,
%U 16435098767896556,0,593753325451468144,0,21450960845508768532,0,774978877336933136632,0
%N Number of n-move knight paths on 8 X 8 board from given corner to same corner.
%H Alois P. Heinz, <a href="/A025600/b025600.txt">Table of n, a(n) for n = 0..400</a>
%F From _Vaclav Kotesovec_, Nov 26 2012: (Start)
%F G.f.: 1 - 2495/3704 - (-2495 + 257062*x^2 - 10940636*x^4 + 261002480*x^6 - 3944912606*x^8 + 40234628876*x^10 - 286888584304*x^12 + 1458140925208*x^14 - 5325997352347*x^16 + 13961752450926*x^18 - 25982840678332*x^20 + 33572692661080*x^22 - 28997305139008*x^24 + 15706751871616*x^26 - 4743107684352*x^28 + 598878986240*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, 0, 0):
%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, 0, 0]; 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_