%I #19 Aug 09 2015 23:43:30
%S 0,0,0,0,0,18,0,852,0,32974,0,1216736,0,44265306,0,1602980908,0,
%T 57958913510,0,2094514154552,0,75677759126258,0,2734176449182980,0,
%U 98781516222538110,0,3568795954184265360,0,128933769847371450506,0,4658126510414596025052,0
%N Number of n-move knight paths on 8x8 board from given corner to adjacent corner.
%H Alois P. Heinz, <a href="/A025602/b025602.txt">Table of n, a(n) for n = 0..400</a>
%F From _Vaclav Kotesovec_, Nov 26 2012: (Start)
%F G.f.: -(2*x^5*(-9 + 186*x^2 - 1433*x^4 + 5172*x^6 - 9228*x^8 + 8048*x^10 - 3360*x^12 + 768*x^14))/((-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))
%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, 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, 0, 7]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, May 28 2015, after _Alois P. Heinz_ *)
%K nonn
%O 0,6
%A _David W. Wilson_