A006748 Number of diagonally symmetric polyominoes with n cells. (Formerly M0366)

0, 0, 1, 0, 2, 2, 7, 5, 26, 22, 91, 79, 326, 301, 1186, 1117, 4352, 4212, 16119, 15849, 60174, 60089, 226146, 228426, 854803, 872404, 3247207, 3342579, 12389106, 12850662, 47448984, 49544820, 182338754, 191529007, 702807040, 742163178, 2716205709, 2882119756

Offset

1,5

Comments

This sequence counts polyominoes with exactly the symmetry group of order 2 generated by a single reflection about a diagonal.

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

John Mason, Table of n, a(n) for n = 1..66 (terms up to a(47) from Robert A. Russell; terms up to a(66) using data from Russell and Shirakawa)

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...

Toshihiro Shirakawa, Harmonic Magic Square, pp 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.

Toshihiro Shirakawa, Enumeration of Polyominoes up to Size N=59, arXiv:2510.22446 [math.CO], 2025.

Formula

a(n) = (A346800(n) - A142886(n)) / 2 - A056878(n). - Robert A. Russell, Aug 25 2021

Crossrefs

Cf. A001168, A346800.

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554.

Sequence in context: A223000 A058625 A300126 * A193548 A396928 A131049

Adjacent sequences: A006745 A006746 A006747 * A006749 A006750 A006751

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Extended to n=28 by Tomás Oliveira e Silva

Status

approved