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Number of board-pair-pile hexagonal polyominoes with n cells.
1

%I #9 Aug 31 2024 15:12:40

%S 1,3,11,44,186,814,3648,16611,76437,354112,1647344,7682237,35873310,

%T 167625690,783470179,3662035980,17115684065,79986841677,373759118882,

%U 1746296080947

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

%H Moa Apagodu (formerly Mohamud Mohammed) and Stirling Chow, <a href="https://arxiv.org/abs/math/0202295">Counting hexagonal lattice animals confined to a strip</a>, arXiv:math/0202295v5 [math.CO], 2009. See Section 7.

%H Moa Apagodu, <a href="http://www.math.rutgers.edu/umbralm1.html">Counting Hexagonal Lattice Animals Using Umbral-Transfer-Matrix Method</a> [dead link]

%H Moa Apagodu, <a href="https://web.archive.org/web/20160910051723/https://www.people.vcu.edu/~mapagodu/programs.html">Maple programs</a>.

%e 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:

%e _ _

%e _/ \ / \_

%e / \_/ \_/ \

%e \_/ \_/

%e / \_ _/ \

%e \_/ \ / \_/

%e \_/ \_/

%e 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:

%e _ _ _ _

%e _/ \ / \_ _/ \ / \_

%e / \_/ \_/ \ / \_/ \_/ \

%e \_/ \_/ \_/ \_/

%e / \_ _/ \ / \_ _/ \

%e \_/ \ / \_/ \_/ \_ _/ \_/

%e / \_/ \_/ \ \_/ \ / \_/

%e \_/ \_/ \_/ \_/

%e / \_ _/ \ _/ \ / \_

%e \_/ \ / \_/ / \_/ \_/ \

%e \_/ \_/ \_/ \_/

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

%Y Cf. A001170, A001207, A059716, A157608.

%K hard,more,nonn

%O 1,2

%A _Moa Apagodu_, Mar 22 2002 and Oct 31 2002