A038119 - OEIS

A038119

Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).
(Formerly M1760)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

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:

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

Table of n, a(n) for n=1..22.

Bert Dobbelaere, Peter Kagey, Drake Thomas, and Andrés R. Vindas-Meléndez, Building with Blocks: Enumerating Polyforms on Tilings, arXiv:2602.23301 [math.CO], 2026. See pp. 8-9.

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)

Wikimedia, Illustration of the 23 pentacubes Illustration of the 112 hexacubes (2025)

FORMULA

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

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 *)

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).