cp's OEIS Frontend

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

%I A000162 M1845 N0731 #101 Jul 03 2025 17:44:58
%S A000162 1,1,2,8,29,166,1023,6922,48311,346543,2522522,18598427,138462649,
%T A000162 1039496297,7859514470,59795121480,457409613979,3516009200564,
%U A000162 27144143923583,210375361379518,1636229771639924,12766882202755783
%N A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.
%C A000162 Here two polycubes that differ by reflection are considered different. - _Joerg Arndt_, Apr 26 2023
%C A000162 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
%D A000162 C. J. Bouwkamp, personal communication.
%D A000162 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.
%D A000162 W. F. Lunnon, personal communication.
%D A000162 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000162 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000162 Nina Bohlmann and Ralf Benölken, <a href="https://doi.org/10.3390/math8101780">Complex Tasks: Potentials and Pitfalls</a>, Mathematics (2020) Vol. 8, No. 10, 1780.
%H A000162 C. J. Bouwkamp, A. J. W. Duijvestijn, & N. J. A. Sloane, <a href="/A000162/a000162.pdf">Correspondence, 1971</a>.
%H A000162 A. Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/Polycubes.html">Polycubes</a>.
%H A000162 A. Clarke, <a href="/A000162/a000162.gif">The 8 tetracubes</a>.
%H A000162 Stanley Dodds, <a href="/A000162/a000162.cs.txt">C# program for this sequence</a>.
%H A000162 Kevin L. Gong, <a href="http://kevingong.com/Polyominoes/Enumeration.html">Polyominoes Home Page</a>.
%H A000162 M. Keller, <a href="http://www.solitairelaboratory.com/polyenum.html">Counting polyforms</a>.
%H A000162 David A. Klarner, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-1/klarner.pdf">Some results concerning polyominoes</a>, Fibonacci Quarterly 3 (1965), 9-20.
%H A000162 Jeffrey R. Long and R. H. Holm, <a href="https://doi.org/10.1021/ja00101a020">Enumeration and structural classification of clusters derived from parent solids ...,</a> J. Amer. Chem. Soc., 116 (1994), 9987-10002.
%H A000162 John Mason, <a href="/A038119/a038119_2.pdf">Coordinate sets of examples of polycube symmetry (version 2)</a>
%H A000162 Phillip Thompson, <a href="https://github.com/philthompson/polycubes-dodds">Rust port of Stanley Dodds's algorithm</a>.
%H A000162 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Polycube.html">Polycube</a>.
%F A000162 a(n) = A066453 + A066454 + A066273 + A066281 + A066283 + A066287 + A066288.
%F A000162 a(n) = 2*A038119 - A007743.
%F A000162 a(n) = A000105 + A006759.
%F A000162 a(n) = A038119(n) + A371397(n) = 2*A371397(n) + A007743(n). - _Robert A. Russell_, Mar 21 2024
%e A000162 Table showing total number and numbers with each group order.
%e A000162 -------------------------------------------------------------
%e A000162 The last 7 columns form sequences A066453, A066454, A066273, A066281, A066283, A066287, A066288.
%e A000162 .n ...A000162 ..group:.1.....2...3...4.6.8.24
%e A000162 .1 .........1..........0.....0...0...0.0.0..1
%e A000162 .2 .........1..........0.....0...0...0.0.1..0
%e A000162 .3 .........2..........0.....1...0...0.0.1..0
%e A000162 .4 .........8..........1.....4...1...0.0.2..0
%e A000162 .5 ........29.........17....10...0...0.0.2..0
%e A000162 .6 .......166........127....34...0...3.1.1..0
%e A000162 .7 ......1023........941....71...4...5.0.1..1
%e A000162 .8 ......6922.......6662...246...0..11.0.2..1
%e A000162 .9 .....48311......47771...522...3..11.0.4..0
%e A000162 10 ....346543.....344708..1783..24..24.2.2..0
%e A000162 11 ...2522522....2518713..3765...4..35.0.5..0
%e A000162 12 ..18598427...18585455.12858..18..84.5.7..0
%e A000162 13 .138462649..138434899.27496.151..92.2.8..1
%e A000162 14 1039496297.1039401564.94525..25.174.4.5..0
%Y A000162 Cf. A001931, A066453.
%Y A000162 Cf. A038119 (unoriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
%K A000162 nonn,nice,hard,more
%O A000162 1,3
%A A000162 _N. J. A. Sloane_ and _J. H. Conway_
%E A000162 The old value for a(11), 2522572, was corrected by _Achim Flammenkamp_ to 2522522, Feb 15 1999.
%E A000162 a(13)-a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27 2001
%E A000162 a(15)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
%E A000162 a(17)-a(20) from _Stanley Dodds_, Dec 11 2023
%E A000162 a(21)-a(22) (using Dodds's algorithm) from _Phillip Thompson_, Feb 07 2024