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

Author

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

Keywords

• nonn
• nice
• hard
• more

Comments

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

Examples

			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
		

References

• 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).

Links

• 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.

Crossrefs

Cf. A001931, A066453.

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

Formula

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

Extensions

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