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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).
4
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
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
See A360055 for the free case.
Sequence in context: A337646 A014303 A076060 * A272340 A090982 A153517
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