A131482 -id:A131482 - cp's OEIS frontend

A038142 Number of planar cata-polyhexes with n cells.

1, 1, 2, 5, 12, 36, 118, 411, 1489, 5572, 21115, 81121, 314075, 1224528, 4799205, 18896981, 74695032, 296275836, 1178741568, 4702507923, 18806505243, 75380203150, 302754225098, 1218239791106

Offset: 1

N. J. A. Sloane

Number of cata-condensed benzenoid hydrocarbons with n hexagons.

Planar cata-polyhexes enumerated by a(n) are the n-celled (planar) polyhexes with perimeter 4n+2, which is the maximal perimeter of an n-celled polyhex. These are such polyhexes that have a tree as their connectedness graph (vertices of this graph correspond to cells and two vertices are connected if the corresponding cells have a common edge). - Tanya Khovanova, Jul 27 2007

			Differs from A002216 starting from a(6) = 36 = A002216(6) - 1: the polyhexes counted by a(6) do not include the ring-like configuration of 6 hexagons where one pair of hexagons which are adjacent from the planar point of view actually have an overlapping pair of external edges rather than a single shared edge. That non-planar configuration is shown in Fig. 2 of the Harary & Read (1970) reference in A002216.

N. Trinajstić, S. Nikolić, J. V. Knop, W. R. Müller and K. Szymanski, Computational Chemical Graph Theory: Characterization, Enumeration, and Generation of Chemical Structures by Computer Methods, Ellis Horwood, 1991.

A. T. Balaban, J. Brunvoll, B. N. Cyvin and S. J. Cyvin, Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures, Tetrahedron, 44(1), 221-228 (1998). See Table 1.
Gunnar Brinkmann, Gilles Caporossi and Pierre Hansen, A Survey and New Results on Computer Enumeration of Polyhex and Fusene Hydrocarbons, J. Chem. Inf. Comput. Sci., 43 (2003), 842-851.
Gilles Caporossi and Pierre Hansen, Enumeration of Polyhex Hydrocarbons to h = 21, J. Chem. Inf. Comput. Sci., 38 (1998), 610-619.
Andrew Clarke, Isoperimetrical Polyhexes
Wenchen He and Wenjie He, Generation and enumeration of planar polycyclic aromatic hydrocarbons, Tetrahedron 42.19 (1986): 5291-5299. See Table 3.
J. V. Knop et al., On the total number of polyhexes, Match, No. 16 (1984), 119-134.
Ratko Tošić, Dragan Mašulović, Ivan Stojmenović, Jon Brunvoll, Bjorg N. Cyvin and Sven J. Cyvin, Enumeration of polyhex hydrocarbons to h = 17, J. Chem. Inf. Comput. Sci., 35 (1995), 181-187.
N. Trinajstich, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski, Computer Generation of Isomeric Structures, Pure & Appl. Chem., Vol. 55, No. 2, pp. 379-390, 1983.
Eric Weisstein's World of Mathematics, Polyhex.
Eric Weisstein's World of Mathematics, Fusene.

Cf. A018190, A038143, A002216.
a(n) <= A000228(n), a(n) <= A057779(2n+1).
A131482 is the analog for polyominoes.

a(n) = A003104(n) + A323851(n). - Andrey Zabolotskiy, Feb 15 2023

a(11) from Tanya Khovanova, Jul 27 2007
a(12)-a(14) from John Mason, May 13 2021
a(15) from Trinajstić et al. (Table 4.2) added by Andrey Zabolotskiy, Feb 08 2023
a(16)-a(17) from Tošić et al., a(18)-a(20) from Caporossi & Hansen and a(21)-a(24) from Brinkmann, Caporossi & Hansen added by Andrey Zabolotskiy, Apr 11 2025

A342243 Triangle T(n,p) read by rows: the number of n-celled polyominoes with perimeter 2p, 2 <= p <= 1+n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 0, 1, 11, 0, 0, 0, 1, 7, 27, 0, 0, 0, 0, 4, 21, 83, 0, 0, 0, 0, 2, 21, 91, 255, 0, 0, 0, 0, 1, 9, 89, 339, 847, 0, 0, 0, 0, 0, 6, 67, 393, 1360, 2829, 0, 0, 0, 0, 0, 1, 45, 325, 1713, 5255, 9734, 0, 0, 0, 0, 0, 1, 23, 275

Offset: 1

R. J. Mathar, Mar 07 2021

			The triangle has rows n=1,2,3,... and columns p=2,3,4,5,...:
  1;
  0, 1;
  0, 0, 2;
  0, 0, 1, 4;
  0, 0, 0, 1, 11;
  0, 0, 0, 1,  7, 27;
  0, 0, 0, 0,  4, 21, 83;
  0, 0, 0, 0,  2, 21, 91, 255;
  0, 0, 0, 0,  1,  9, 89, 339,  847;
  0, 0, 0, 0,  0,  6, 67, 393, 1360, 2829;
  0, 0, 0, 0,  0,  1, 45, 325, 1713, 5255, 9734;
  ...

John Mason, Table of T(n,k) for n <= 18 (n <= 17 from R. J. Mathar)
Andrew Clarke, Isoperimetrical Polyominoes
John Mason, Incomplete rows for n>18

Cf. A000105 (row sums), A057730 (column sums), A131482 (diagonal), A131487 (skew antidiagonal sums), A027709 (number of leading zeros per row), A100092 (first nonzero in each row).

A131487(e) = Sum_{e=2*n+p} T(n,p).

A304196 Number of free tree-like polycubes of size n in three dimensions.

Original entry on oeis.org

1, 1, 2, 6, 21, 91, 484, 2817, 17788, 116741, 788081, 5414701, 37703459, 265182187

Offset: 1

Lotfi Bouallagui, May 07 2018

A free tree-like polycube of size n in three dimensions is a face-connected set of n cells in the cubic lattice, up to translation, rotation and reflection, whose dual graph has no cycles.

Alexandre Blondin Massé, Julien de Carufel, Alain Goupil, and Maxime Samson, Fully-leafed tree-like polyominoes and polycubes, In Combinatorial algorithms, volume 10765 of Lecture Notes of Computer Science, 28th International workshop, IWOCA 2017, Newcastle, NSW, Australia, Springer, 2018.

Cf. A118356 (fixed tree-like 3d-polycubes), A304198 (free tree-like 4d-polycubes), A131482 (free tree-like polyominoes).

Terms a(13) and a(14) from Joerg Arndt and Márk Péter Légrádi, May 21 2023

A131487 a(n) is the number of polyominoes with n edges, including inner edges.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 4, 0, 1, 11, 1, 7, 27, 4, 21, 85, 21, 92, 264, 89, 345, 914, 394, 1405, 3155, 1736, 5530, 11400, 7586, 22022, 41756, 32702, 87158, 156412, 139253, 346836, 592661, 589101, 1379837, 2275935, 2476770, 5501846, 8830267, 10363627, 21970992, 34594887, 43188260, 87950618

Offset: 1

Tanya Khovanova, Jul 28 2007

An n-celled polyomino with perimeter p has (4n+p)/2 edges. The maximum number of edges in an n-celled polyomino is 3n+1.

			A single cell has 4 edges; a domino has 7 edges (this includes the edge between the two cells); both trominoes have 10 edges; their possible orientations are not considered distinct. Thus a(4) = a(7) = 1, a(10) = 2, and a(n) = 0 for n < 10 not equal to 4 or 7.
a(22) = 85 = 83 + 2: there are 83 polyominoes with 7 cells and perimeter 16 (such as a 1 X 7 strip) and two polyominoes with 8 cells and perimeter 12 (a 3 X 3 square without a corner and a 4 X 2 rectangle), and each of these polyominoes has 22 edges.
a(23) = 21. a(24) = 91+1. a(25) = 255+9. a(26) = 89. a(27) = 339+6. a(28) = 847+67. a(34) = 9734+1655+11. a(35) = 7412+174. - _R. J. Mathar_, Feb 22 2021

Andrew Clarke, Isoperimetrical Polyominoes

Cf. A000105, A057730, A342243.

Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2), A131488 (analog for hexagonal tiling).

See A342243 for formula.

a(23)-a(35) from R. J. Mathar, Feb 22 2021
a(36)-a(39) from R. J. Mathar, Mar 11 2021
a(40)-a(44) from R. J. Mathar, Mar 24 2021
a(45)-a(54) from John Mason, Apr 28 2023

A304195 Number of fully-leafed free tree-like polyominoes of size n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 12, 3, 1, 6, 74, 11, 2, 21, 408, 40, 4, 76, 2053, 148, 11, 279

Offset: 1

Lotfi Bouallagui, May 07 2018

A free tree-like polyomino of size n is a connected set of n cells in the square lattice, up to translation, rotation and reflection, whose dual graph has no cycles. It is called fully-leafed when it has the maximal number of leaves over all the same sized free tree-like polyominoes.

			a(5) = 1:
.  #
. ###
.  #
a(6) = 2:
.  #   .  #
. #### . ####
.  #   .   #
a(7) = 12:
. # # . # #  .  # #  .    #  .  #    .   #
. ### . #### . ##### . ##### . ##### . #####
. # # . #    .       .  #    .  #    .  #
.
.  # # .  # # .  #   .  #   .  #   .   #
. #### . #### .  #   . ##   . ##   . #####
.  #   .   #  . #### .  ### .  ### .   #
.      .      .  #   .  #   .   #  .

Alexandre Blondin Massé, Julien de Carufel, Alain Goupil, and Maxime Samson. Fully-leafed tree-like polyominoes and polycubes. In Combinatorial algorithms, volume 10765 of Lecture Notes of Computer Science, 28th International workshop, IWOCA 2017, Newcastle, NSW, Australia, Springer, 2018.

Cf. A131482 (free tree-like polyominoes), A304197, A304199 (fully-leafed free tree-like polycubes in 3 and 4 dimensions resp.).

A304198 Number of free tree-like 4d-polycubes of size n.

Original entry on oeis.org

1, 1, 2, 6, 24, 122, 838, 6759, 61600, 600875, 6139448

Offset: 1

Lotfi Bouallagui, May 07 2018

A free tree-like polycube of size n in four dimensions is a hyperface-connected set of n cells in the 4d-hypercubic lattice, up to translation, rotation and reflection, whose dual graph has no cycles.

Alexandre Blondin Massé, Julien de Carufel, Alain Goupil, and Maxime Samson, Fully-leafed tree-like polyominoes and polycubes, In Combinatorial algorithms, volume 10765 of Lecture Notes of Computer Science, 28th International workshop, IWOCA 2017, Newcastle, NSW, Australia, Springer, 2018.

Cf. A191094 (fixed tree-like 4d-polycubes), A131482 (free tree-like polyominoes), A304196 (free tree-like 3d-polycubes).

A130622 Number of polyominoes with perimeter at most 2n.

Original entry on oeis.org

0, 1, 2, 5, 11, 36, 122, 538, 2526, 13166, 71153, 400109, 2300430, 13504780, 80547558, 487327904

Offset: 1

Tanya Khovanova, Aug 10 2007

The perimeter of a polyomino is always even.
a(n) is partial sums of A057730.
a(n+1) >= A130866(n).
Is there another prime term beyond {2, 5, 11}?

Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2).
Cf. A057730 (number of polyominoes (A000105) with perimeter 2n).
Cf. A130866 (number of polyominoes with at most n cells).

Offset corrected by John Mason, Jan 16 2023

a(9)-a(16) from John Mason, Jan 16 2023

A135942 Number of n-celled polyominoes with perimeter < 2n+2.

Original entry on oeis.org

0, 0, 0, 1, 1, 8, 25, 114, 438, 1826, 7339, 29876, 120346, 485155, 1947974, 7812084, 31267765

Offset: 1

Tanya Khovanova, Mar 03 2008

These are n-cell polyominoes with the perimeter less than the maximum possible one.

If we associate a graph to a polyomino with vertices representing cells and edges representing two cells with a common edge, then this sequence enumerates polyominoes whose corresponding graphs have cycles.

a(n) = A000105(n) - A131482(n).

More terms from Joerg Arndt, May 21 2018

A350030 a(n) is the number of n-celled unholey treelike polyominoes.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 82, 250, 815, 2685, 9072, 30889, 106290, 367733, 1279573, 4470304, 15676760, 55147665

Offset: 1

John Mason, Dec 09 2021

Cf. A308300 (number of simply connected square animals with n cells and k internal vertices, triangle read by rows).

Cf. A131482 (polyominoes with perimeter 2n+2, i.e., treelike).

a(n) = A308300(n,0).

A359522 Number of free, holey, treelike polyominoes of n cells.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 5, 32, 144, 662, 2835, 11955, 49083, 199029, 796867, 3163384, 12463807

Offset: 1

John Mason, Jan 04 2023

			a(7) = 1 because of the holey heptomino.

Cf. A000105, A001419, A131482.

cp's OEIS Frontend

A038142 Number of planar cata-polyhexes with n cells.

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A342243 Triangle T(n,p) read by rows: the number of n-celled polyominoes with perimeter 2p, 2 <= p <= 1+n.

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A304196 Number of free tree-like polycubes of size n in three dimensions.

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A131487 a(n) is the number of polyominoes with n edges, including inner edges.

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A304195 Number of fully-leafed free tree-like polyominoes of size n.

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A304198 Number of free tree-like 4d-polycubes of size n.

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A130622 Number of polyominoes with perimeter at most 2n.

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A135942 Number of n-celled polyominoes with perimeter < 2n+2.

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A350030 a(n) is the number of n-celled unholey treelike polyominoes.

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A359522 Number of free, holey, treelike polyominoes of n cells.

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