Gill Barequet - cp's OEIS frontend

A196593 Number of fixed tree-like convex polyominoes.

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.

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).

A191098 Number of n-cell fixed tree-like polycubes in 8 dimensions.

Original entry on oeis.org

1, 8, 120, 2248, 47636, 1088017, 26424957

Offset: 1

Gill Barequet, May 25 2011

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.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A066158 (fixed tree-like polyominoes), A118356, A191094, A191095, A191096, A191097 (fixed tree-like polycubes in 3, 4, 5, 6, and 7 dimensions, resp.).

A191097 Number of n-cell fixed tree-like polycubes in 7 dimensions.

Original entry on oeis.org

1, 7, 91, 1463, 26460, 516691, 10654378

Offset: 1

Gill Barequet, May 25 2011

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.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A066158 (fixed tree-like polyominoes), A118356, A191094, A191095, A191096, A191098 (fixed tree-like polycubes in 3, 4, 5, 6, and 8 dimensions, resp.).

A191096 Number of n-cell fixed tree-like polycubes in 6 dimensions.

Original entry on oeis.org

1, 6, 66, 886, 13281, 213978, 3630090, 64012932

Offset: 1

Gill Barequet, May 25 2011

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.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A066158 (fixed tree-like polyominoes), A118356, A191094, A191095, A191097, A191098 (fixed tree-like polycubes in 3, 4, 5, 7, and 8 dimensions, resp.).

A191095 Number of n-cell fixed tree-like polycubes in 5 dimensions.

Original entry on oeis.org

1, 5, 45, 485, 5775, 73437, 979335, 13536225, 192393410, 2796392165

Offset: 1

Gill Barequet, May 25 2011

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.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A066158 (fixed tree-like polyominoes), A118356, A191094, A191096, A191097, A191098 (fixed tree-like polycubes in 3, 4, 6, 7, and 8 dimensions, resp.).

A191094 Number of n-cell fixed tree-like polycubes in 4 dimensions.

Original entry on oeis.org

1, 4, 28, 228, 2018, 18892, 184400, 1857856, 19189675, 202214452

Offset: 1

Gill Barequet, May 25 2011

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.

Gill Barequet, Gil Ben-Shachar, Martha Carolina Osegueda, Applications of Concatenation Arguments to Polyominoes and Polycubes, EuroCG '20, 36th European Workshop on Computational Geometry, (Würzburg, Germany, 16-18 March 2020).

Cf. A066158 (fixed tree-like polyominoes), A118356, A191095, A191096, A191097, A191098 (fixed tree-like polycubes in 3, 5, 6, 7, and 8 dimensions, resp.).

A191092 Number of n-cell polycubes that are proper in n-3 dimensions.

Original entry on oeis.org

0, 1, 61, 2836, 129288, 6160640, 313921008, 17239040000, 1021644763392, 65244849242112, 4477975127425280, 329252714454974464, 25850313756000000000, 2160223055912342913024, 191558954408834121740288, 17973564914103712921681920

Offset: 3

Gill Barequet, May 25 2011

A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-cell polycubes in n-3 dimensions, Proc. 17th Ann. Int. Computing and Combinatorics Conference, Dallas, TX, Lecture Notes in Computer Science, 6842, Springer-Verlag, 180-191, August 2011.
Gill Barequet, Solomon W. Golomb, and David A. Klarner, Polyominoes. (This is a revision, by G. Barequet, of the chapter of the same title originally written by the late D. A. Klarner for the first edition, and revised by the late S. W. Golomb for the second edition.) Preprint, 2016, http://www.csun.edu/~ctoth/Handbook/chap14.pdf
G. Barequet, M. Shalah, Automatic Proofs for Formulae Enumerating Proper Polycubes, 31st International Symposium on Computational Geometry (SoCG'15). Editors: Lars Arge and János Pach; pp. 19-22, 2015.
R. Barequet, G. Barequet, and G. Rote, Formulae and growth rates of high-dimensional polycubes, Combinatorica, 30 (2010), 257-275.

Vincenzo Librandi, Table of n, a(n) for n = 3..100
A. Asinowski, G. Barequet, R. Barequet, and G. Rote, Proper n-Cell Polycubes in n-3 Dimensions, J. Int. Seq. 15 (2012) #12.8.4.
M. Shalah, Formulae and growth rates of animals on cubical and triangular lattices, PhD Thesis, Israel Inst. Techn. (2017)

Cf. A127670, A171860.

Diagonal 3 of A195739.

[2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3: n in [3..40]]; // Vincenzo Librandi, May 26 2011

a[n_]:=2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3 ; Array[a, 40, 3] (* Stefano Spezia, Sep 09 2018 *)

a(n)=2^(n-6)*n^(n-7)*(n-3)*(12*n^5-104*n^4+360*n^3-679*n^2+1122*n-1560)/3

a(n) = 2^(n-6)*n^(n-7)*(n-3)*(12*n^5 - 104*n^4 + 360*n^3 - 679*n^2 + 1122*n - 1560)/3.

cp's OEIS Frontend

User: Gill Barequet

A196593 Number of fixed tree-like convex polyominoes.

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

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A191098 Number of n-cell fixed tree-like polycubes in 8 dimensions.

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A191097 Number of n-cell fixed tree-like polycubes in 7 dimensions.

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A191096 Number of n-cell fixed tree-like polycubes in 6 dimensions.

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A191095 Number of n-cell fixed tree-like polycubes in 5 dimensions.

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A191094 Number of n-cell fixed tree-like polycubes in 4 dimensions.

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A191092 Number of n-cell polycubes that are proper in n-3 dimensions.

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