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
Keywords
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.
Programs
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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 *)
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
Extensions
Definition corrected at the suggestion of Emeric Deutsch, Mar 03 2007
More terms from Vaclav Kotesovec, Jun 06 2021
Comments