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

1, 8, 168, 5120, 190120, 8039808, 373369920, 18576523680, 972362837160, 52832252432960, 2950644716576128, 168192125309339040, 9735527029198105408, 570163460613978204800, 33697054064651581144800, 2005939326990647575285920, 120109818840839172931095720

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

Cf. A094061, A253974, A254129, A307347.

Sequence in context: A039699 A307347 A253974 * A254129 A359926 A334780

Adjacent sequences: A254456 A254457 A254458 * A254460 A254461 A254462

Keyword

nonn

Author

Vaclav Kotesovec, Jan 30 2015

Status

approved