A038119 Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).

1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• hard
• more
• nice

Comments

a(1)-a(12) computed by Achim Flammenkamp.

A000162 but with one copy of each mirror-image deleted.

From R. J. Mathar, Mar 19 2018: (Start)

We can split the numbers into an irregular table which lists in row n how many configurations have c contacts for c >= 0:

1;

0 1;

0 0 2;

0 0 0 6 1;

0 0 0 0 21 2;

0 0 0 0 0 91 19 2;

0 0 0 0 0 0 484 110 12 1;

0 0 0 0 0 0 0 2817 852 129 12 0 1;

0 0 0 0 0 0 0 0 17788 6321 1166 132 5 1;

Row lengths are 1+A007818(n). Row sums are a(n).

(End)

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

References

• S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.
• 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. [See https://books.google.nl/books?id=ja7iBQAAQBAJ&pg=PA101]
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Achim Flammenkamp, Home page.
• Kevin L. Gong, Polyominoes Home Page.
• John Mason, Counting free polycubes.
• John Mason, Coordinate sets of examples of polycube symmetry (version 2)

Crossrefs

A038119 = (A007743+A000162)/2, A007743 = 2*A038119 - A000162, A000162 = 2*A038119 - A007743.

32nd row of A366766.

Cf. A000162 (oriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).

Cf. for each symmetry: A376964, A376965, A376966, A376967, A376968, A376969, A376970, A376972, A376973, A376974, A376975, A376976, A376977, A376978, A376979, A376980, A376981, A376982, A376983, A377127, A376984, A376985, A376986, A376987, A376988, A376989, A377128, A376990, A376991, A377129, A377130, A377131, A376971

Programs

Mathematica

A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {, }][[All, 2]]];
A000162 = A@000162;
A007743 = A@007743;
a[n_] := (A007743[[n]] + A000162[[n]])/2;
a /@ Range[16] (* Jean-François Alcover, Jan 16 2020 *)

Formula

a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - Robert A. Russell, Mar 21 2024

Extensions

More terms from Brendan Owen (brendan_owen(AT)yahoo.com), Jan 02 2002

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

More terms from John Mason, Sep 19 2024