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

Author

N. J. A. Sloane, Sep 03 2000

Keywords

• nonn

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

Examples

			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
		

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

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.

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

Extensions

More terms from Robert A. Russell, Jan 16 2019