A001170 -id:A001170 - cp's OEIS frontend

A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

1, 3, 11, 44, 186, 814, 3648, 16611, 76437, 354112, 1647344, 7682237, 35873310, 167625690, 783470179, 3662035980, 17115684065, 79986841677, 373759118882, 1746296080947

Offset: 1

Moa Apagodu, Mar 22 2002 and Oct 31 2002

			This sequence first diverges from A059716 at n = 4. a(4) is 2 greater than A059716(4) because a(4) counts the following 2 fixed polyhexes of 4 cells that contain a column with two contiguous blocks of cells:
     _    _
   _/ \  / \_
  / \_/  \_/ \
  \_/      \_/
  / \_    _/ \
  \_/ \  / \_/
    \_/  \_/
This sequence first diverges from A001207 at n = 7. a(7) is 4 less than A001207(7) because a(7) does not count the following 4 fixed polyhexes of 7 cells that contain a column with more than two contiguous blocks of cells:
     _    _        _        _
   _/ \  / \_    _/ \      / \_
  / \_/  \_/ \  / \_/      \_/ \
  \_/      \_/  \_/          \_/
  / \_    _/ \  / \_        _/ \
  \_/ \  / \_/  \_/ \_    _/ \_/
  / \_/  \_/ \    \_/ \  / \_/
  \_/      \_/      \_/  \_/
  / \_    _/ \     _/ \  / \_
  \_/ \  / \_/    / \_/  \_/ \
    \_/  \_/      \_/      \_/

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Section 7.
Moa Apagodu, Counting Hexagonal Lattice Animals Using Umbral-Transfer-Matrix Method [dead link]
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A157608.

The sequence is generated by a Maple program that accompanies the paper "Counting Hexagonal Lattice Animals using Umbral-Transfer-Matrix Method (UTMM)"

A157608 Array read by antidiagonals, giving number of fixed hexagonal polyominoes of height up to n/2 and with hexagonal cell count k.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 3, 2, 0, 0, 1, 3, 6, 2, 0, 0, 1, 3, 10, 11, 2, 0, 0, 1, 3, 11, 25, 19, 2, 0, 0, 1, 3, 11, 37, 61, 32, 2, 0, 0, 1, 3, 11, 43, 111, 142, 53, 2, 0, 0, 1, 3, 11, 44, 153, 320, 323, 87, 2, 0, 0, 1, 3, 11, 44, 177, 514, 896, 723, 142, 2, 0, 0

Offset: 1

Jonathan Vos Post, Mar 02 2009

			The array begins:
================================================
n=1 | 0 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=2 | 1 | 0 |  0 |  0 |   0 |   0 |   0 |    0 |
n=3 | 1 | 2 |  2 |  2 |   2 |   2 |   2 |    2 |
n=4 | 1 | 3 |  6 | 11 |  19 |  32 |  53 |   87 |
n=5 | 1 | 3 | 10 | 25 |  61 | 142 | 323 |  723 |
n=6 | 1 | 3 | 11 | 37 | 111 | 320 | 896 | 2461 |
================================================

Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, Counting hexagonal lattice animals confined to a strip, arXiv:math/0202295v5 [math.CO], 2009. See Table 1.
Moa Apagodu, Maple programs.

Cf. A001170, A001207, A059716, A068091, A308359.

T(4, 4) = 11:
                                               
   / \/ \ / \/ \  / \         / \   / \   / \
  / \/ \/ \/ \/ \ \/ \/ \ / \/ \/ / \/ \ / \/ \
  \/ \/     \/ \/   \/ \/ \/ \/  / \/ \/ \/ \/ \
                          \/     \/    \/             \/
                              _
   / \  / \/ \  / \/ \   / \  / \
  / \/ \ \/ \/  \/ \/ / \/  \/ \_
  \/ \/   \/ \  / \/  / \/ \  / \/ \
    \/       \/  \/    \/ \/  \/ \_/

T(n, k) = A001207(k) for n >= 2*k. - Andrey Zabolotskiy, Aug 31 2024

Definition not clear to me! "Height" refers to the lattice or to the polyominoes? - N. J. A. Sloane, Mar 14 2009

Name clarified and more terms added by Andrey Zabolotskiy, Aug 24 2024

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A068091 Number of board-pair-pile hexagonal polyominoes with n cells.

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