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A038119
Number of n-celled solid polyominoes (or free polycubes, allowing mirror-image identification).
(Formerly M1760)
68
1, 1, 2, 7, 23, 112, 607, 3811, 25413, 178083, 1279537, 9371094, 69513546, 520878101, 3934285874, 29915913663, 228779330204, 1758309223457, 13573319825615, 105192814197984, 818136047201932, 6383528588447574
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:
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).
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 *)
KEYWORD
nonn,hard,more,nice
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
STATUS
approved