A038119 - OEIS

%I M1760 #53 Oct 18 2024 10:49:48

%S 1,1,2,7,23,112,607,3811,25413,178083,1279537,9371094,69513546,

%T 520878101,3934285874,29915913663,228779330204,1758309223457,

%U 13573319825615,105192814197984,818136047201932,6383528588447574

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

%C a(1)-a(12) computed by _Achim Flammenkamp_.

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

%C From _R. J. Mathar_, Mar 19 2018: (Start)

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

%C   1;

%C   0 1;

%C   0 0 2;

%C   0 0 0 6 1;

%C   0 0 0 0 21 2;

%C   0 0 0 0 0 91 19 2;

%C   0 0 0 0 0 0 484 110 12 1;

%C   0 0 0 0 0 0 0 2817 852 129 12 0 1;

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

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

%C (End)

%C 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

%D S. W. Golomb, Polyominoes. Scribner's, NY, 1965; second edition (Polyominoes: Puzzles, Packings, Problems and Patterns) Princeton Univ. Press, 1994.

%D 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]

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/cgi-bin/cgiwrap/achim/index.cgi">Home page</a>.

%H Kevin L. Gong, <a href="http://kevingong.com/Polyominoes/Enumeration.html">Polyominoes Home Page</a>.

%H John Mason, <a href="/A038119/a038119_1.pdf">Counting free polycubes</a>.

%H John Mason, <a href="/A038119/a038119_2.pdf">Coordinate sets of examples of polycube symmetry (version 2)</a>

%F a(n) = A000162(n) - A371397(n) = A371397(n) + A007743(n). - _Robert A. Russell_, Mar 21 2024

%t A[s_Integer] := With[{s6 = StringPadLeft[ToString[s], 6, "0"]}, Cases[ Import["https://oeis.org/A" <> s6 <> "/b" <> s6 <> ".txt", "Table"], {_, _}][[All, 2]]];

%t A000162 = A@000162;

%t A007743 = A@007743;

%t a[n_] := (A007743[[n]] + A000162[[n]])/2;

%t a /@ Range[16] (* _Jean-François Alcover_, Jan 16 2020 *)

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

%Y 32nd row of A366766.

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

%Y 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

%K nonn,hard,more,nice

%O 1,3

%A _N. J. A. Sloane_

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

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

%E More terms from _John Mason_, Sep 19 2024