A056877 Number of polyominoes with n cells, symmetric about two orthogonal axes.

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

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), 191-203.

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