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%I #51 Nov 01 2020 06:08:16
%S 1,8,168,5120,190120,7964208,362370624,17532536736,889716433320,
%T 46887220540160,2546408317827088,141659449976239104,
%U 8033749056463329472,462687411167492828000,26980019699392099317600,1589091557661690119997120,94361786346423775855372200
%N Number of 2n-move closed giraffe paths on an unbounded chessboard from a given square to the same square.
%C Giraffe is a (fairy chess) leaper [1,4].
%C Conjecture: Number of 2n-move closed paths of leaper [r,s] on an unbounded chessboard, where 0 < r < s and gcd(r,s)=1, is asymptotic to 2^(6*n+1) / ((r^2+s^2)*Pi*n) if r+s is even, and 2^(6*n) / ((r^2+s^2)*Pi*n) if r+s is odd.
%H Vaclav Kotesovec, <a href="/A253974/b253974.txt">Table of n, a(n) for n = 0..552</a>
%H Vaclav Kotesovec, <a href="/A253974/a253974.txt">Conjectured recurrence (of order 8)</a>
%H Vaclav Kotesovec, <a href="/A253974/a253974.jpg">Examples of closed giraffe paths</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Fairy_chess_piece">Fairy chess piece</a>
%F a(n) ~ 64^n / (17*Pi*n).
%F a(n) = the constant term in the expansion of (x*y^4 + x^4*y + 1/x*y^4 + 1/x^4*y + x/y^4 + x^4/y + 1/x/y^4 + 1/x^4/y)^(2*n). - _Vaclav Kotesovec_, Apr 01 2019
%p b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>5*n, 0,
%p `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 1],
%p [1, 4], [-4, 1], [-1, 4], [4, -1], [1, -4], [-4, -1], [-1, -4]])))
%p end:
%p a:= n-> b(2*n, 0$2):
%p seq(a(n), n=0..25); # after _Alois P. Heinz_
%p # second Maple program:
%p poly:=expand((x*y^4+x^4*y+y^4/x+y/x^4+x/y^4+x^4/y+1/(x*y^4)+1/(x^4*y))^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do: # _Vaclav Kotesovec_, Apr 03 2019
%t b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 4n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{4, 1}, {1, 4}, {-4, 1}, {-1, 4}, {4, -1}, {1, -4}, {-4, -1}, {-1, -4}}}]]];
%t a[n_] := b[2n, 0, 0];
%t a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 01 2020, after Maple *)
%Y Cf. A094061, A254129, A254459, A307347.
%K nonn
%O 0,2
%A _Vaclav Kotesovec_, Jan 31 2015
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