A066158 -id:A066158 - cp's OEIS frontend

A019441 Coefficients in generating function for radius of gyration of the sequence A066158.

1, 2, 28, 244, 1680, 10214, 57476, 305476, 1553632, 7641218, 36608932, 171666468, 790650724, 3586822020, 16062938368, 71135451440, 311964025352, 1356392904818, 5852609697844, 25081266854732

Offset: 1

N. J. A. Sloane, Nov 05 2008

nonn

I. Jensen, Table of n, a(n) for n = 1..42 [From the arXiv paper]
I. Jensen, Enumerations of lattice animals and trees, J. Stat. Phys. 103 (3-4) (2001) 865-881, Table II.
I. Jensen, Enumerations of lattice animals and trees, arXiv:cond-mat/0007239.

Cf. A066158, A056261.

A192074 a(n) = A066158(n)-2 with a(0)=1.

Original entry on oeis.org

0, 0, 4, 16, 53, 172, 568, 1906, 6471, 22200, 76884, 268350, 942649, 3329606, 11817580, 42120338, 150682448, 540832272, 1946892840, 7027047846, 25424079337, 92185846606, 334925007126, 1219054432488, 4444545298877, 16229462702150, 59347661054362

Offset: 1

N. J. A. Sloane, Jun 22 2011

nonn

Gadi Aleksandrowicz and Gill Barequet, Parallel Enumeration of Lattice Animals, in Frontiers in Algorithmics and Algorithmic Aspects in Information and Management, Lecture Notes in Computer Science, 2011, Volume 6681/2011, 90-99. See the table of A_2^(PT)(n).

Cf. A066158, A006762, A192075, A192076, A192077.

A118356 Number of clusters with n vertices, n-1 edges and zero contacts on the simple cubic lattice.

Original entry on oeis.org

1, 3, 15, 83, 486, 2967, 18748, 121725, 807381, 5447203, 37264974, 257896500, 1802312605, 12701190885, 90157130289, 644022007040, 4626159163233

Offset: 1

R. J. Mathar, May 14 2006

a(n)<=A001931(n) due to the "no-contact" restriction.

An alternative wording for a(n) is the number of n-cell fixed tree-like polycubes in 3 dimensions. - 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).
N. Madras, C. E. Soteros, S. G. Whittington, J. L. Martin et al., The free energy of a collapsing branched polymer, J Phys A: Math Gen 23 (1990) 5327-5350

Cf. A066158 (fixed tree-like polyominoes), A191094, A191095, A191096, A191097, A191098 (fixed tree-like polycubes in 4, 5, 6, 7, and 8 dimensions, resp.).

a(1)=1 added by Gill Barequet, May 25 2011

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.).

A061157 Numbers k such that k*2^m-1 is prime for exactly one exponent m in the range 0<=m<=k.

Original entry on oeis.org

11, 22, 37, 74, 109, 151, 247, 253, 337, 382, 403, 506, 509, 547, 674, 739, 764, 779, 806, 1325, 1478, 1528, 1558, 1589, 1597, 1612, 1871, 1987, 2263, 2279, 2317, 2759, 2797, 3056, 3116, 3194, 3329, 3341, 3719, 3742, 3761, 3821, 4001, 4412, 4526, 4558, 4559, 4603, 4736, 4813

Offset: 1

Patrick De Geest, Apr 15 2001

nonn

Michael S. Branicky, Table of n, a(n) for n = 1..125

Cf. A061153, A061154, A061155, A061156. See A066158 for exponents.

Offset corrected and more terms from Sean A. Irvine, Jan 24 2023

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.).

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.).

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.).

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.).

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.

cp's OEIS Frontend

A019441 Coefficients in generating function for radius of gyration of the sequence A066158.

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A192074 a(n) = A066158(n)-2 with a(0)=1.

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A118356 Number of clusters with n vertices, n-1 edges and zero contacts on the simple cubic lattice.

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

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A061157 Numbers k such that k*2^m-1 is prime for exactly one exponent m in the range 0<=m<=k.

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

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

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

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

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A196593 Number of fixed tree-like convex polyominoes.

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