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, 457409613979, 3516009200564, 27144143923583, 210375361379518, 1636229771639924, 12766882202755783

Offset

1,3

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

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..22.

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.

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.

John Mason, Coordinate sets of examples of polycube symmetry

Phillip Thompson, Rust port of Stanley Dodds's algorithm.

Eric Weisstein's World of Mathematics, Polycube.

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

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, A066453.

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

Sequence in context: A185033 A287045 A009419 * A052437 A131318 A010749

Adjacent sequences: A000159 A000160 A000161 * A000163 A000164 A000165

Keyword

nonn,nice,hard,more,changed

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)-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

Status

approved