A067675 -id:A067675 - cp's OEIS frontend

A067676 Number of fixed directed convex polyominoes with n cells.

1, 2, 5, 13, 33, 82, 200, 481, 1144, 2699, 6329, 14775, 34381, 79819, 185001, 428290, 990716, 2290424, 5293153, 12229209, 28249088, 65246630, 150687282, 347993954, 803620981, 1855754764, 4285319033, 9895581541, 22850547145, 52765494456, 121843455307

Offset: 1

Steven Finch, Feb 04 2002

nonn

Sean A. Irvine, Table of n, a(n) for n = 1..500
M. Bousquet-Mélou and J.-M. Fédou, The generating function of convex polyominoes: the resolution of a q-differential system, Discrete Math. 137 (1995) 53-75.
Sergi Elizalde, Symmetric peaks and symmetric valleys in Dyck paths, arXiv:2008.05669 [math.CO], 2020.

Cf. A006958, A067675 (fixed convex polyominoes), A191148 (fixed line-convex polycubes in 3 dimensions).

More terms from Sean A. Irvine, Jan 02 2024

A191148 Number of n-cell fixed line-convex polycubes in 3 dimensions.

Original entry on oeis.org

1, 3, 15, 86, 522, 3241, 20256, 126520

Offset: 1

Gill Barequet, May 26 2011

A polycube is "line-convex" if every axis-parallel line intersects it in at most one continuous sequence of cells.

G. Aleksandrowicz and G. Barequet, Parallel enumeration of lattice animals, Proc. 5th Int. Frontiers of Algorithmics Workshop, Zhejiang, China, Lecture Notes in Computer Science, 6681, Springer-Verlag, 90-99, May 2011.

Cf. A067675 (convex polyominoes).

A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

Original entry on oeis.org

3, 2, 0, 5, 5, 6, 9, 4, 3, 0, 4, 0, 0, 5, 9, 0, 3, 1, 1, 7, 0, 2, 0, 2, 8, 6, 1, 7, 7, 8, 3, 8, 2, 3, 4, 2, 6, 3, 7, 7, 1, 0, 8, 9, 1, 9, 5, 9, 7, 6, 9, 9, 4, 4, 0, 4, 7, 0, 5, 5, 2, 2, 0, 3, 5, 5, 1, 8, 3, 4, 7, 9, 0, 3, 5, 9, 1, 6, 7, 4, 6, 9, 1, 7, 6, 4, 1, 8, 2, 6, 9, 5, 7, 8, 0, 5, 2, 5, 0, 7, 8, 4, 9, 9

Offset: 1

Jean-François Alcover, Sep 03 2014

Essentially the same digit sequence as A137421. - R. J. Mathar, Sep 06 2014

			3.20556943040059031170202861778382342637710891959769944...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 48.

Cf. A001169, A006958, A067675, A246772.

v = Root[x^3 - 5*x^2 + 7*x - 4, x, 1]; RealDigits[v, 10, 104] // First

v = first root of x^3 - 5*x^2 + 7*x - 4 = (x-2)^3+(x-2)^2-(x-2)-2.

A001169(n) ~ u*v^n, where u = A246772.

A196593 Number of fixed tree-like convex polyominoes.

Original entry on oeis.org

1, 2, 6, 18, 51, 134, 328, 758, 1677, 3594, 7530, 15530, 31687, 64190, 129420, 260142, 521889, 1045730, 2093806, 4190402, 8384091, 16772022, 33548496, 67102118, 134210101, 268426874, 536861298, 1073731098, 2147471727, 4294954094, 8589920020, 17179853150

Offset: 1

Gill Barequet, Oct 04 2011

In a 1-1 mapping with permutations that avoid the patterns (1423, 4213, 2314, 2431, 2413, <3142,{2},{2}>) (the fourth pattern is bivincular).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Gadi Aleksandrowicz, Andrei Asinowski and Gill Barequet, A polyominoes-permutations injection and tree-like convex polyominoes, Journal of Combinatorial Theory, Series A, Volume 119, Issue 3, April 2012, Pages 503-520
A. Goupil, H. Cloutier, and F. Nouboud, Enumeration of inscribed polyominos, FPSCA 2010 (San Francisco) DMTS proc. AN 2010, 737-748, eq. (10)
Index entries for linear recurrences with constant coefficients, signature (6,-14,16,-9,2).

Cf. A001168 (fixed polyominoes), A066158 (fixed tree polyominoes), A067675 (fixed convex polyominoes).

LinearRecurrence[{6,-14,16,-9,2},{1,2,6,18,51},50] (* Harvey P. Dale, Oct 16 2011 *)

G.f.: (x*(1-4*x+8*x^2-6*x^3+4*x^4))/((1-x)^4*(1-2*x)).
a(n) = 6*a(n-1) - 14*a(n-2) + 16*a(n-3) - 9*a(n-4) + 2*a(n-5).
a(n) = 2^(n+2) - (n^3-n^2+10*n+4)/2.

A246772 Decimal expansion of 'u', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

Original entry on oeis.org

1, 8, 0, 9, 1, 5, 5, 0, 1, 8, 8, 1, 5, 6, 0, 6, 0, 9, 5, 1, 5, 8, 9, 5, 7, 7, 3, 0, 1, 0, 0, 0, 1, 8, 0, 0, 4, 9, 4, 4, 2, 9, 1, 0, 3, 3, 9, 9, 8, 8, 1, 0, 0, 0, 4, 9, 9, 5, 9, 4, 8, 3, 2, 4, 4, 3, 8, 9, 8, 1, 7, 8, 0, 8, 2, 4, 5, 6, 3, 2, 8, 6, 5, 8, 4, 2, 9, 4, 6, 2, 4, 4, 0, 7, 4, 9, 0, 4, 9, 1, 1, 5, 5

Offset: 0

Jean-François Alcover, Sep 03 2014

			0.180915501881560609515895773010001800494429103399881...

Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 48.

Cf. A001169, A006958, A067675, A246773.

u = Root[944*x^3 - 295*x^2 + 28*x - 1, x, 1]; RealDigits[u, 10, 103] // First

u = first root of 944*x^3 - 295*x^2 + 28*x - 1.

A001169(n) ~ u*v^n, where v = A246773.

A359661 a(n) is the number of free convex polyominoes of n cells.

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 72, 191, 478, 1211, 2973, 7274, 17455, 41645, 98271, 230848, 539000, 1254936

Offset: 1

John Mason, Jan 10 2023

A convex polyomino is such that any vertical or horizontal line connecting two points within the polyomino remains completely within the polyomino.
It has a perimeter of length equal to that of its enclosing rectangle.
A polyomino is convex if and only if (i) it is a board-pile polyomino and (ii) rotated 90 degrees it is still a board-pile-polyomino.

Cf. A000105, A001169, A067675.

cp's OEIS Frontend

A067676 Number of fixed directed convex polyominoes with n cells.

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A191148 Number of n-cell fixed line-convex polycubes in 3 dimensions.

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A246773 Decimal expansion of 'v', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

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A196593 Number of fixed tree-like convex polyominoes.

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A246772 Decimal expansion of 'u', an auxiliary constant associated with the asymptotic number of row-convex polyominoes.

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Programs

Mathematica

Formula

A359661 a(n) is the number of free convex polyominoes of n cells.

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