A000228 -id:A000228 - cp's OEIS frontend

A103464 Number of polyominoes consisting of n regular unit heptagons.

1, 1, 2, 7, 25, 118, 558, 2876, 14982, 80075, 431889, 2354991, 12930257, 71459124, 396978189, 2215609864

Offset: 3

Sascha Kurz, Jun 09 2006

Matthias Koch and Sascha Kurz, Enumeration of generalized polyominoes, arXiv:math/0605144 [math.CO], 2006.

Cf. A000577, A000104, A000228, A103466, A103467, A103468, A103469, A103470, A103471, A103472, A103473, A120102, A120103, A120104.

A137193 Number of n-celled polyhexes with perimeter < 4n+2.

Original entry on oeis.org

0, 0, 1, 2, 10, 46, 215, 1037, 5083, 24918, 122437

Offset: 1

Tanya Khovanova, Mar 03 2008

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

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

A000228(n)-A038142(n)

A176638 Partial sums of A018190.

Original entry on oeis.org

1, 2, 5, 12, 34, 115, 446, 1881, 8386, 38472, 179701, 849285, 4047541, 19415118, 93623028, 453486806, 2205081449, 10758731196, 52651373968, 258365785954, 1270930958357, 6265738653554, 30952863554094, 153191072337297

Offset: 1

Jonathan Vos Post, Apr 22 2010

nonn

Partial sums of number of planar simply-connected polyhexes (or benzenoid hydrocarbons) with n hexagons. The only known primes in the partial sum are 2 and 5.

			a(6) = 1 + 1 + 3 + 7 + 22 + 81 = 115.

Cf. A000228, A002216, A018190, A038142-A038147.

a(n) = SUM[i=1..n] A018190(i).

A330211 Number of free pentagonal polyforms with n cells on the order-4 pentagonal tiling of the hyperbolic plane.

Original entry on oeis.org

1, 1, 1, 2, 8, 28, 143, 747, 4346, 25974, 160869, 1015723, 6531611, 42592880

Offset: 0

Peter Kagey, Mar 05 2020

The order-4 pentagonal tiling of the hyperbolic plane has Schläfli symbol {5,4}.

This sequence is computed from via program by Christian Sievers in the Code Golf Stack Exchange link.

Code Golf Stack Exchange, Impress Donald Knuth by counting polyominoes on the hyperbolic plane
Wikipedia, Order-4 pentagonal tiling
Wikipedia, Polyform

Analogs with different Schläfli symbols are A000105 ({4,4}), A000207 ({3,oo}), A000228 ({6,3}), A000577 ({3,6}), A005036 ({4,oo}), A119611 ({4,5}), A330659 ({3,7}), A332930 ({4,6}), and A333018 ({7,3}).

C
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// See the Code Golf link.
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GAP
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bc
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a(8)-a(13) from Ed Wynn, Feb 16 2021

A342963 a(n) is the number of sticky polyhexes with 2*n cells.

Original entry on oeis.org

1, 2, 15, 110, 1051, 10636, 113290, 1234189, 13674761, 153289285

Offset: 1

Woosuk Kwak, Mar 31 2021

A sticky polyhex is defined as follows:
- A single dihex (polyhex of size 2) is a sticky polyhex.
- If a polyhex X is sticky, X plus a dihex Y is also sticky if X and Y share at least two unit sides.
- Any polyhex that cannot be formed by the above definition is not sticky.
This sequence counts free polyhexes; two polyhexes which are equivalent under reflection and/or rotation are counted only once.
a(n) < A000228(2n) for n > 1.

			The two sticky tetrahexes are:
    * *    * * *
     * *    *
The following is the full list of 15 sticky hexahexes (polyhexes of size 6):
    * * *    * * *    *        * * * *    * * *
     * *      * *    * * * *    * *          * * *
      *          *        *
---
    * *       *        * *       * *     * * *
     * * *   * * * *    * * *   * * *       * *
        *       *      *         *         *
---
    * * *    * * *    * * * *   * * *    * * *
       * *    * * *    *   *       *        * *
          *                       * *        *

Woosuk Kwak, Sticky Polyhexes, Code Golf Stack Exchange.

Cf. A000228.

cp's OEIS Frontend

A103464 Number of polyominoes consisting of n regular unit heptagons.

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A137193 Number of n-celled polyhexes with perimeter < 4n+2.

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A176638 Partial sums of A018190.

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A330211 Number of free pentagonal polyforms with n cells on the order-4 pentagonal tiling of the hyperbolic plane.

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C

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A342963 a(n) is the number of sticky polyhexes with 2*n cells.

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