A126764 Number of L-convex polyominoes with n cells, that is, convex polyominoes where any two cells can be connected by a path internal to the polyomino and which has at most 1 change of direction (i.e., one of the four orientations of the letter L).

1, 1, 2, 6, 15, 35, 76, 156, 310, 590, 1098, 1984, 3515, 6094, 10398, 17434, 28837, 47038, 75820, 120794, 190479, 297365, 460056, 705576, 1073473, 1620680, 2429352, 3616580, 5349359, 7863564, 11491946, 16700534, 24140606, 34716813, 49682700, 70766326, 100343410

Offset

0,3

Comments

This sequence counts fixed L-convex polyominoes. See crossrefs for the free case. - Allan C. Wechsler, Jan 27 2023

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, Combinatorial Exploration: An algorithmic framework for enumeration, arXiv:2202.07715 [math.CO], 2022.

G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, Combinatorial aspects of L-convex polyominoes, European J. Combin. 28 (2007), no. 6, 1724-1741.

Anthony Guttmann and Vaclav Kotesovec, L-convex polyominoes and 201-avoiding ascent sequences, arXiv:2109.09928 [math.CO], 2021.

Formula

The reference gives a generating function.

Conjecture: a(n) ~ c * exp(Pi*sqrt(13*n/6)) / n^(3/2), where c = 13*sqrt(2) / 768. - Anthony Guttmann and Vaclav Kotesovec, Jun 09 2021

Mathematica

nmax = 50; f[k_, x_] := f[k, x] = (If[k == 0, 1, If[k == 1, 1 + 2*x - x^2, Normal[Series[2*f[k-1, x] - (1 - x^k)^2 * f[k-2, x], {x, 0, nmax}]]]]); CoefficientList[Series[1 + Sum[x^k * f[k-1, x]/((Product[(1 - x^j)^2, {j, 1, k-1}] * (1 - x^k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 06 2021 *)

Crossrefs

Cf. A000105, A003480, A126765.

See A360055 for the free case.

Sequence in context: A337646 A014303 A076060 * A272340 A090982 A153517

Adjacent sequences: A126761 A126762 A126763 * A126765 A126766 A126767

Keyword

nonn

Author

N. J. A. Sloane, based on email from Simone Rinaldi (rinaldi(AT)unisi.it), Feb 23 2007

Extensions

Definition corrected at the suggestion of Emeric Deutsch, Mar 03 2007

More terms from Vaclav Kotesovec, Jun 06 2021

Status

approved