A307347 Number of 2n-move closed antelope paths on an unbounded chessboard from a given square to the same square.

1, 8, 168, 5120, 190120, 7939008, 357713664, 17010543264, 842994009000, 43192225007360, 2275378947981568, 122724475613935104, 6753785574641857024, 378138077830110886400, 21486835143540141873120, 1236506847203439155401920, 71934214120446285067176360

Offset

0,2

Comments

Antelope is a leaper [3,4].

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..552

Formula

a(n) = the constant term in the expansion of (x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^(2*n).

Conjecture: a(n) ~ 64^n / (25*Pi*n).

Maple

b:= proc(n, x, y) option remember; `if`(max(x, y)>4*n or x+y>7*n, 0,
      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[4, 3],
      [3, 4], [-4, 3], [-3, 4], [4, -3], [3, -4], [-4, -3], [-3, -4]])))
    end:
a:= n-> b(2*n, 0$2):
seq(a(n), n=0..25);
# second Maple program:
poly := expand((x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2): z:=1: for n to 100 do z:=expand(z*poly): print(n, coeff(coeff(z, x, 0), y, 0)); end do:

Mathematica

poly = Expand[(x^4*y^3 + x^3*y^4 + 1/x^4*y^3 + 1/x^3*y^4 + x^4/y^3 + x^3/y^4 + 1/x^4/y^3 + 1/x^3/y^4)^2]; z = 1; Flatten[{1, Table[z = Expand[z*poly]; z[[1]], {n, 1, 15}]}]

Crossrefs

Cf. A094061, A253974, A254129, A254459.

Sequence in context: A220808 A221022 A039699 * A253974 A254459 A254129

Adjacent sequences: A307344 A307345 A307346 * A307348 A307349 A307350

Keyword

nonn

Author

Vaclav Kotesovec, Apr 03 2019

Status

approved