A006759 -id:A006759 - cp's OEIS frontend

A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480, 457409613979, 3516009200564, 27144143923583, 210375361379518, 1636229771639924, 12766882202755783

Offset: 1

N. J. A. Sloane and J. H. Conway

Here two polycubes that differ by reflection are considered different. - Joerg Arndt, Apr 26 2023

Number of oriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Mar 21 2024

			Table showing total number and numbers with each group order.
-------------------------------------------------------------
The last 7 columns form sequences A066453, A066454, A066273, A066281, A066283, A066287, A066288.
.n ...A000162 ..group:.1.....2...3...4.6.8.24
.1 .........1..........0.....0...0...0.0.0..1
.2 .........1..........0.....0...0...0.0.1..0
.3 .........2..........0.....1...0...0.0.1..0
.4 .........8..........1.....4...1...0.0.2..0
.5 ........29.........17....10...0...0.0.2..0
.6 .......166........127....34...0...3.1.1..0
.7 ......1023........941....71...4...5.0.1..1
.8 ......6922.......6662...246...0..11.0.2..1
.9 .....48311......47771...522...3..11.0.4..0
10 ....346543.....344708..1783..24..24.2.2..0
11 ...2522522....2518713..3765...4..35.0.5..0
12 ..18598427...18585455.12858..18..84.5.7..0
13 .138462649..138434899.27496.151..92.2.8..1
14 1039496297.1039401564.94525..25.174.4.5..0

C. J. Bouwkamp, personal communication.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
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).

Nina Bohlmann and Ralf Benölken, Complex Tasks: Potentials and Pitfalls, Mathematics (2020) Vol. 8, No. 10, 1780.
C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, Correspondence, 1971.
A. Clarke, Polycubes.
A. Clarke, The 8 tetracubes.
Stanley Dodds, C# program for this sequence.
Kevin L. Gong, Polyominoes Home Page.
M. Keller, Counting polyforms.
David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.
Jeffrey R. Long and R. H. Holm, Enumeration and structural classification of clusters derived from parent solids ..., J. Amer. Chem. Soc., 116 (1994), 9987-10002.
John Mason, Coordinate sets of examples of polycube symmetry (version 2)
Phillip Thompson, Rust port of Stanley Dodds's algorithm.
Eric Weisstein's World of Mathematics, Polycube.

Cf. A001931, A066453.

Cf. A038119 (unoriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).

a(n) = A066453 + A066454 + A066273 + A066281 + A066283 + A066287 + A066288.
a(n) = 2*A038119 - A007743.
a(n) = A000105 + A006759.
a(n) = A038119(n) + A371397(n) = 2*A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

The old value for a(11), 2522572, was corrected by Achim Flammenkamp to 2522522, Feb 15 1999.
a(13)-a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001
a(15)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(17)-a(20) from Stanley Dodds, Dec 11 2023
a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024

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

A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.

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