A006758 -id:A006758 - cp's OEIS frontend

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

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

nonn

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, ...

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

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).

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.

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

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

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

A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 6, 3, 0, 1, 17, 17, 4, 0, 1, 59, 131, 52, 7, 0, 1, 195, 915, 709, 153, 13, 0, 1, 703, 6553, 8946, 3350, 454, 28

Offset: 1

N. J. A. Sloane, Sep 22 2011

From Petros Hadjicostas, Jan 11 2019: (Start)
Table 1 (p. 366) in Lunnon (1975) contains more terms. Because the table there (in the reference) has incomplete columns, the extra terms do not appear in this triangular sequence (array).
Entry DR(n=11, d=2) in Table 1 (p. 366) must be a typo. It should not be 33890, but 33895. This was corrected by N. J. A. Sloane in 2011 in the documentation of sequence A006758. (See also sequence A000988.)
(End)
The number of oriented polyominoes (chiral pairs counted as two) here is the sum of the number of unoriented polyominoes (chiral pairs counted as one) in A049430 and the number of chiral pairs. - Robert A. Russell, May 03 2020

			Triangle begins:
n\d| 0    1    2    3    4    5    6    7
---+---------------------------------=---
1  | 1
2  | 0    1
3  | 0    1    1
4  | 0    1    6    3
5  | 0    1   17   17    4
6  | 0    1   59  131   52    7
7  | 0    1  195  915  709  153   13
8  | 0    1  703 6553 8946 3350  454   28
...

W. F. Lunnon, Counting multidimensional polyominoes, Computer Journal 18 (4) (1975) 366-367.

Columns give A006758 (and A000988), A006759, A006760, A006761.
Cf. A195739, A049430.
Cf. A000055, A045649, A036364, A036365.

From Robert A. Russell, May 03 2020: (Start)
For n > 1, DR(n,n-1) = A000055(n) + A045649(n).
DR(n,n-2) = A036364(n) + A036365(n).
We can add unoriented and chiral pairs for the top two diagonals. The summands have quick algorithms. (End)

Sequence corrected by Petros Hadjicostas, Jan 11 2019 after observation by Jon E. Schoenfield

cp's OEIS Frontend

A195741 Erroneous version of A006758.

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A000988 Number of one-sided polyominoes with n cells.

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A195738 Triangle read by rows: DR(n,d) is the number of properly d-dimensional polyominoes with n cells, modulo translations and rotations (n >= 1, 0 <= d <= n-1).

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