

A056877


Number of polyominoes with n cells, symmetric about two orthogonal axes.


27



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

1,6


COMMENTS

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


LINKS

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), 191203.
D. H. Redelmeier, Table 3 of Counting polyominoes...
Wikipedia, Symmetries of polyominoes


FORMULA

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


EXAMPLE

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
.
XXXX
XXXX
.
XX
XXXX
XX
.
X X
XXXX
X X


CROSSREFS

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


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Sep 03 2000


EXTENSIONS

More terms from Robert A. Russell, Jan 16 2019


STATUS

approved



