A006749 Number of asymmetric polyominoes with n cells. (Formerly M3906)

0, 0, 0, 1, 5, 20, 84, 316, 1196, 4461, 16750, 62878, 237394, 899265, 3422111, 13069026, 50091095, 192583152, 742560511, 2870523142, 11122817672, 43191285751, 168046076423, 654997492842, 2557223459805, 9999080270766, 39152997087077, 153511067364760

Offset

1,5

Comments

This sequence counts polyominoes whose symmetry group has order 1.

References

A. R. Conway and A. J. Guttmann, On two-dimensional percolation, J. Phys. A: Math. Gen. 28(1995) 891-904.

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..59 (using data from Robert A. Russell and Shirakawa)

Tomás Oliveira e Silva, Enumeration of polyominoes

T. R. Parkin, L. J. Lander, and D. R. Parkin, Polyomino Enumeration Results, presented at SIAM Fall Meeting, 1967, and accompanying letter from T. J. Lander (annotated scanned copy). See page 21.

D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.

D. H. Redelmeier, Table 3 of Counting polyominoes...

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

Formula

a(n) + A259090(n) = A000105(n). - R. J. Mathar, Sep 29 2021

a(n) = A000105(n) - A006746(n) - A006748(n) - A006747(n) - A056877(n) - A056878(n) - A144553(n) - A142886(n). - John Mason, Jan 07 2026

Crossrefs

Cf. A001168 (fixed).

Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A259090.

Sequence in context: A292760 A270322 A056884 * A002213 A099949 A006231

Adjacent sequences: A006746 A006747 A006748 * A006750 A006751 A006752

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

Extended to n=28 by Tomás Oliveira e Silva.

Status

approved