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A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.

%I A254459 #32 Nov 01 2020 06:08:07
%S A254459 1,8,168,5120,190120,8039808,373369920,18576523680,972362837160,
%T A254459 52832252432960,2950644716576128,168192125309339040,
%U A254459 9735527029198105408,570163460613978204800,33697054064651581144800,2005939326990647575285920,120109818840839172931095720
%N A254459 Number of 2n-move closed zebra paths on an unbounded chessboard from a given square to the same square.
%C A254459 Zebra is a (fairy chess) leaper [2,3].
%C A254459 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 A254459 Vaclav Kotesovec, <a href="/A254459/b254459.txt">Table of n, a(n) for n = 0..542</a>
%H A254459 Vaclav Kotesovec, <a href="/A254459/a254459.jpg">Examples of closed zebra paths</a>
%H A254459 Vaclav Kotesovec, <a href="/A254459/a254459.txt">Conjectured recurrence (of order 6)</a>
%H A254459 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fairy_chess_piece">Fairy chess piece</a>
%F A254459 a(n) ~ 64^n / (13*Pi*n).
%F A254459 a(n) = the constant term in the expansion of (x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^(2*n). - _Vaclav Kotesovec_, Apr 01 2019
%p A254459 b:= proc(n, x, y) option remember; `if`(max(x, y)>3*n or x+y>5*n, 0,
%p A254459       `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[3, 2],
%p A254459       [2, 3], [-3, 2], [-2, 3], [3, -2], [2, -3], [-3, -2], [-2, -3]])))
%p A254459     end:
%p A254459 a:= n-> b(2*n, 0$2):
%p A254459 seq(a(n), n=0..25); # after _Alois P. Heinz_
%p A254459 # second Maple program:
%p A254459 poly:=expand((x^2*y^3 + x^3*y^2 + 1/x^2*y^3 + 1/x^3*y^2 + x^2/y^3 + x^3/y^2 + 1/x^2/y^3 + 1/x^3/y^2)^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 A254459 b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 3n || x + y > 5n, 0, If[n == 0, 1, Sum[b[n - 1, Abs[x + l[[1]]], Abs[y + l[[2]]]], {l, {{3, 2}, {2, 3}, {-3, 2}, {-2, 3}, {3, -2}, {2, -3}, {-3, -2}, {-2, -3}}}]]];
%t A254459 a[n_] := b[2n, 0, 0];
%t A254459 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 01 2020, after Maple *)
%Y A254459 Cf. A094061, A253974, A254129, A307347.
%K A254459 nonn
%O A254459 0,2
%A A254459 _Vaclav Kotesovec_, Jan 30 2015