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A056877
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Number of polyominoes with n cells, symmetric about two orthogonal axes.
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27
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0, 1, 1, 1, 1, 2, 3, 4, 4, 8, 10, 15, 17, 30, 35, 60, 64, 117, 128, 236, 241, 459, 476, 937, 912, 1813, 1789, 3706, 3456, 7187, 6779, 14712, 13161, 28571, 25839, 58457, 50348, 113798, 98957, 232718, 193375, 453969, 380522, 927601, 745248, 1813219, 1468202, 3702063
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OFFSET
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1,6
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COMMENTS
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This sequence counts polyominoes with exactly the symmetry group D_4 generated by horizontal and vertical reflections.
The subset of (2n)-ominoes with edge centers in this set are enumerated by A346799(n). - Robert A. Russell, Dec 15 2021
Polyominoes centered about square centers and vertices are enumerated by A351190 and A351191 respectively. - John Mason, Feb 15 2022
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LINKS
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Robert A. Russell, Table of n, a(n) for n = 1..81
Tomás Oliveira e Silva, Enumeration of polyominoes
D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Wikipedia, Symmetries of polyominoes
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FORMULA
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a(n) = A351190(n) + A346799(n/2) + A351191(n/4) if we accept the convention that Axxxxxx(y) = 0 for any noninteger y. - John Mason, Feb 15 2022
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EXAMPLE
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For a(8)=4, the four octominoes with exactly fourfold symmetry and axes of symmetry parallel to the edges of the cells are a row of eight cells, two adjacent rows of four cells, a row of four cells with another four cells adjacent to its center cells, and a row of four cells with another four cells adjacent to its end cells (but not in the original row):
XXXXXXXX
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XXXX
XXXX
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XX
XXXX
XX
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X X
XXXX
X X
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CROSSREFS
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Cf. A000105, A001168, A006746, A006748, A056878, A006747, A006749, A346799, A351190, A351191.
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.
Sequence in context: A330147 A241037 A097093 * A267232 A269606 A269640
Adjacent sequences: A056874 A056875 A056876 * A056878 A056879 A056880
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Sep 03 2000
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EXTENSIONS
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More terms from Robert A. Russell, Jan 16 2019
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STATUS
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approved
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