OFFSET
0,2
COMMENTS
Zebra is a (fairy chess) leaper [2,3].
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.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..542
Vaclav Kotesovec, Examples of closed zebra paths
Vaclav Kotesovec, Conjectured recurrence (of order 6)
Wikipedia, Fairy chess piece
FORMULA
a(n) ~ 64^n / (13*Pi*n).
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
MAPLE
b:= proc(n, x, y) option remember; `if`(max(x, y)>3*n or x+y>5*n, 0,
`if`(n=0, 1, add(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]])))
end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25); # after Alois P. Heinz
# second Maple program:
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
MATHEMATICA
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}}}]]];
a[n_] := b[2n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 01 2020, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Jan 30 2015
STATUS
approved