A059713 Number of multi-directed animals on the square lattice.

1, 2, 6, 19, 63, 214, 738, 2571, 9020, 31806, 112572, 399548, 1421145, 5063254, 18062902, 64505148, 230547424, 824547052, 2950565215, 10562978104, 37828967047, 135516406962, 485586082540, 1740321129477, 6238304153594, 22364826278804, 80189282398662

Offset

1,2

Comments

This class of animals is in bijection with a simple class of heaps of dimers. It generalizes directed animals.

Links

Table of n, a(n) for n=1..27.

M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers, Discrete Math. 258 (2002), no. 1-3, 235-274.

M. Bousquet-Mélou and A. Rechnitzer, Lattice animals and heaps of dimers (preprint)

Formula

The generating function is known in closed form. Closed, but big. It is proved to be non-D-finite.

Mathematica

n = 30;
q = (1-x-Sqrt[(1+x)(1-3x)]) / (2x) + O[x]^(n+1);
gf = q / ((1-q)(1-Sum[q^(k+1)/(1-q^k (1+q)), {k, n}]));
CoefficientList[gf, x] (* Andrei Zabolotskii, Nov 24 2025 *)

Crossrefs

Directed animals: A005773. Triangular lattice: A059715.

Sequence in context: A120900 A284216 A059712 * A109262 A006724 A057409

Adjacent sequences: A059710 A059711 A059712 * A059714 A059715 A059716

Keyword

nonn

Author

Mireille Bousquet-Mélou, Feb 08 2001

Extensions

Terms a(21) onwards from Andrei Zabolotskii, Nov 24 2025

Status

approved