A007743 Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

1, 1, 2, 6, 17, 58, 191, 700, 2515, 9623, 36552, 143761, 564443, 2259905, 9057278, 36705846, 149046429, 609246350, 2495727647, 10267016450, 42322763940, 174974139365

Offset: 1

Author

Arlin Anderson (starship1(AT)gmail.com)

Keywords

• nonn
• nice

Comments

A000162 but with both copies of each mirror-image deleted.

An achiral polyomino is identical to its reflection. Many of these achiral polyominoes do not have a plane of symmetry. For example, the hexomino with cell centers (0,0,0), (0,0,1), (0,1,1), (1,1,1), (1,2,1), and (1,2,2) has a center of symmetry at (1/2,1,1) but no plane of symmetry. The decomino with cell centers (0,0,0), (0,0,1), (0,1,1), (0,2,1), (0,2,2), (1,0,2), (1,1,2), (1,1,1), (1,1,0), and (1,2,0) has no plane or center of symmetry. - Robert A. Russell, Mar 21 2024

Links

• G. Thimm, Emails to N. J. A. Sloane, Sep. 1994

Crossrefs

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.

Cf. A000162 (oriented), A038119 (unoriented), A371397 (chiral), A001931 (fixed).

Component symmetries: A376969, A376970, A376971, A376972, A376973, A376974, A376977, A376978, A376979, A376980, A376981, A376983, A376984, A376985, A376986, A376987, A376988, A376989, A376990, A376991, A377129, A377130.

Formula

a(n) = A000162(n) - 2*A371397(n) = A038119(n) - A371397(n). - Robert A. Russell, Mar 21 2024

Extensions

a(13)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

Changed "symmetric" to "mirror-symmetric" in the title by George Sicherman, Feb 21 2018

Changed "mirror-symmetric" to "achiral" in the title to ensure that a plane of symmetry is not required. - Robert A. Russell, Mar 21 2024

a(17)-a(22) from John Mason, Sep 19 2024