A000988 Number of one-sided polyominoes with n cells.

1, 1, 1, 2, 7, 18, 60, 196, 704, 2500, 9189, 33896, 126759, 476270, 1802312, 6849777, 26152418, 100203194, 385221143, 1485200848, 5741256764, 22245940545, 86383382827, 336093325058, 1309998125640, 5114451441106, 19998172734786, 78306011677182, 307022182222506, 1205243866707468, 4736694001644862

Offset: 0

Author

N. J. A. Sloane, hugh(AT)mimosa.com (D. Hugh Redelmeier)

Keywords

• nonn

Comments

A000105(n) + A030228(n) = a(n) because the number of free polyominoes plus the number of polyominoes lacking bilateral symmetry equals the number of one-sided polyominoes. - Graeme McRae, Jan 05 2006

Names for the first few polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, enneomino (aka nonomino), decomino, hendecomino (aka undecomino), dodecomino, ...

Examples

			a(0) = 1 as there is 1 empty polyomino with #cells = 0. - _Fred Lunnon_, Jun 24 2020
		

References

• S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
• J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 229.
• W. F. Lunnon, personal communication.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• 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 = 0..50 (terms 0..45,47,49 from Toshihiro Shirakawa).
• W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18(4) (1975), 366-367.
• Ed Pegg, Jr., Illustrations of polyforms.
• Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [Broken link]
• Jaime Rangel-Mondragon, Polyominoes and Related Families, The Mathematica Journal, 9(3) (2005), 609-640. [From the internet archive]
• D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete Math., 36 (1981), 191-203.
• Toshihiro Shirakawa, Harmonic Magic Square, pp. 3-4: Enumeration of Polyominoes considering the symmetry, April 2012.
• Eric Weisstein's World of Mathematics, Polyomino.
• Wikipedia, Polyomino.

Crossrefs

See A006758 for another version. Subtracting 1 gives first column of A195738. Cf. A000105 (unoriented), A030228 (chiral), A030227 (achiral), A001168 (fixed).

Formula

a(n) = 2*A006749(n) + A006746(n) + A006748(n) + 2*A006747(n) + A056877(n) + A056878(n) + 2*A144553(n) + A142886(n). - Andrew Howroyd, Dec 04 2018

a(n) = 2*A000105(n) - A030227(n) = 2*A030228(n) + A030227(n). - Robert A. Russell, Feb 03 2022

Extensions

a(0) = 1 added by N. J. A. Sloane, Jun 24 2020