A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells. (Formerly M1845 N0731)

1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480

Offset

1,3

Comments

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

References

C. J. Bouwkamp, personal communication.

J. R. Long and R. H. Holm, Enumeration and structural classification of clusters derived from parent solids ..., J. Amer. Chem. Soc., 116 (1984), 9987-10002.

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

Table of n, a(n) for n=1..16.

Nina Bohlmann and Ralf Benölken, Complex Tasks: Potentials and Pitfalls, Mathematics (2020) Vol. 8, No. 10, 1780.

C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971

A. Clarke, Polycubes

A. Clarke, The 8 tetracubes.

Kevin L. Gong, Polyominoes Home Page.

M. Keller, Counting polyforms

David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.

Eric Weisstein's World of Mathematics, Polycube

Formula

a(n) = A066273 + A066281 + A066283 + A066287 + A066288 + A066453 + A066545.

a(n) = 2*A038119 - A007743.

a(n) = A000105 + A006759.

Example

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

Crossrefs

Cf. A001931, A038119, A007743, A066453.

Sequence in context: A185033 A287045 A009419 * A052437 A131318 A010749

Adjacent sequences: A000159 A000160 A000161 * A000163 A000164 A000165

Keyword

nonn,nice,hard,more

Author

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

Extensions

The old value for a(11), 2522572, was corrected by Achim Flammenkamp to 2522522, Feb 15 1999.

a(13) and a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27, 2001

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

Link updated by William Rex Marshall, Dec 16 2009

Status

approved