A126764 - OEIS

%I #42 Mar 22 2024 18:45:15

%S 1,1,2,6,15,35,76,156,310,590,1098,1984,3515,6094,10398,17434,28837,

%T 47038,75820,120794,190479,297365,460056,705576,1073473,1620680,

%U 2429352,3616580,5349359,7863564,11491946,16700534,24140606,34716813,49682700,70766326,100343410

%N 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).

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

%H Vaclav Kotesovec, <a href="/A126764/b126764.txt">Table of n, a(n) for n = 0..2000</a>

%H Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, and Henning Ulfarsson, <a href="https://arxiv.org/abs/2202.07715">Combinatorial Exploration: An algorithmic framework for enumeration</a>, arXiv:2202.07715 [math.CO], 2022.

%H G. Castiglione, A. Frosini, E. Munarini, A. Restivo and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.ejc.2006.06.020">Combinatorial aspects of L-convex polyominoes</a>, European J. Combin. 28 (2007), no. 6, 1724-1741.

%H Anthony Guttmann and Vaclav Kotesovec, <a href="https://arxiv.org/abs/2109.09928">L-convex polyominoes and 201-avoiding ascent sequences</a>, arXiv:2109.09928 [math.CO], 2021.

%F The reference gives a generating function.

%F 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

%t 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 *)

%Y Cf. A000105, A003480, A126765.

%Y See A360055 for the free case.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, based on email from Simone Rinaldi (rinaldi(AT)unisi.it), Feb 23 2007

%E Definition corrected at the suggestion of _Emeric Deutsch_, Mar 03 2007

%E More terms from _Vaclav Kotesovec_, Jun 06 2021