This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A366766 #18 Oct 30 2023 15:33:19
%S A366766 1,0,1,0,1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,1,1,0,1,0,1,3,2,1,0,1,0,1,7,5,
%T A366766 0,1,0,1,0,1,20,16,0,1,1,0,1,0,1,60,55,0,2,1,1,0,1,0,1,204,222,0,5,2,
%U A366766 2,1,0,1,0,1,702,950,0,12,5,5,0,1
%N A366766 Array read by antidiagonals, where each row is the counting sequence of a certain type of free polyominoids (see comments).
%C A366766 A (D,d)-polyominoid is a connected set of d-dimensional unit cubes (cells) with integer coordinates in D-dimensional space. For normal polyominoids, two cells are connected if they share a (d-1)-dimensional facet, but here we allow connections where the cells share a lower-dimensional face.
%C A366766 Each row is the counting sequence (by number of cells) of (D,d)-polyominoids with certain restrictions on the allowed connections between cells. Two cells have a connection of type (g,h) if they intersect in a (d-g)-dimensional unit cube and extend in d-h common dimensions. For example, d-dimensional polyominoes use connections of type (1,0), polyplets use connections of types (1,0) (edge connections) and (2,0) (corner connections), normal (3,2)-polyominoids use connections of types (1,0) ("soft" connections) and (1,1) ("hard" connections), hard polyominoids use connections of type (1,1).
%C A366766 Each row corresponds to a triple (D,d,C), where 1 <= d <= D and C is a set of pairs (g,h) with 1 <= g <= d and 0 <= h <= min(g, D-d). The k-th term of that row is the number of free k-celled (D,d)-polyominoids with connections of the types in C. Connections of types not in C are permitted, but the polyominoids must be connected through the specified connections only. For example, polyominoes may have cells that intersect in a point (g = 2) and hard polyominoids can have soft connections (h = 0) that are not needed to keep the polyominoids connected.
%C A366766 The rows are sorted first by D, then by d, and finally by a binary vector indicating which types of connections are allowed, where the connection types (g,h) are sorted lexicographically. (See table in cross-references.)
%C A366766 For each pair (D,d), the first row is 1, 0, 0, ..., corresponding to (D,d,{}) (no connections allowed).
%C A366766 The number of rows corresponding to given values of D and d is 2^((d+1)*(d+2)/2-1) if 2*d <= D and 2^((D-d+1)*(3*d-D+2)/2-1) otherwise.
%H A366766 Pontus von Brömssen, <a href="/A366766/b366766.txt">Table of n, a(n) for n = 1..210</a> (first 20 antidiagonals).
%H A366766 Pontus von Brömssen, <a href="/A366766/a366766.py.txt">Python programs that relate row numbers and parameter sets</a>.
%H A366766 Wikipedia, <a href="https://en.wikipedia.org/wiki/Polyominoid">Polyominoid</a>.
%H A366766 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%e A366766 Array begins:
%e A366766 n\k| 1 2 3 4 5 6 7 8 9 10 11 12
%e A366766 ---+------------------------------------------------------------
%e A366766 1 | 1 0 0 0 0 0 0 0 0 0 0 0
%e A366766 2 | 1 1 1 1 1 1 1 1 1 1 1 1
%e A366766 3 | 1 0 0 0 0 0 0 0 0 0 0 0
%e A366766 4 | 1 1 1 1 1 1 1 1 1 1 1 1
%e A366766 5 | 1 1 3 7 20 60 204 702 2526 9180 33989 126713
%e A366766 6 | 1 2 5 16 55 222 950 4265 19591 91678 434005 2073783
%e A366766 7 | 1 0 0 0 0 0 0 0 0 0 0 0
%e A366766 8 | 1 1 2 5 12 35 108 369 1285 4655 17073 63600
%e A366766 9 | 1 1 2 5 12 35 108 369 1285 4655 17073 63600
%e A366766 10 | 1 2 5 22 94 524 3031 18770 118133 758381 4915652 32149296
%e A366766 11 | 1 0 0 0 0 0 0 0 0 0 0 0
%e A366766 12 | 1 1 1 1 1 1 1 1 1 1 1 1
%Y A366766 Cf. A366767 (fixed), A366768.
%Y A366766 The following table lists some sequences that are rows of the array, together with the corresponding values of D, d, and C. Some sequences occur in more than one row. Notation used in the table:
%Y A366766 X: Allowed connection.
%Y A366766 -: Not allowed connection (but may occur "by accident" as long as it is not needed for connectedness).
%Y A366766 .: Not applicable for (D,d) in this row.
%Y A366766 !: d < D and all connections have h = 0, so these polyominoids live in d < D dimensions only.
%Y A366766 *: Whether a connection of type (g,h) is allowed or not is independent of h.
%Y A366766 | | | connections |
%Y A366766 | | | g:1122233334 |
%Y A366766 n | D | d | h:0101201230 | sequence
%Y A366766 ----+---+---+--------------+---------
%Y A366766 1 | 1 | 1 | * -......... | A063524
%Y A366766 2 | 1 | 1 | * X......... | A000012
%Y A366766 3 |!2 | 1 | * --........ | A063524
%Y A366766 4 |!2 | 1 | X-........ | A000012
%Y A366766 5 | 2 | 1 | -X........ | A361625
%Y A366766 6 | 2 | 1 | * XX........ | A019988
%Y A366766 7 | 2 | 2 | * -.-....... | A063524
%Y A366766 8 | 2 | 2 | * X.-....... | A000105
%Y A366766 9 | 2 | 2 | * -.X....... | A000105
%Y A366766 10 | 2 | 2 | * X.X....... | A030222
%Y A366766 11 |!3 | 1 | * --........ | A063524
%Y A366766 12 |!3 | 1 | X-........ | A000012
%Y A366766 13 | 3 | 1 | -X........ | A365654
%Y A366766 14 | 3 | 1 | * XX........ | A365559
%Y A366766 15 |!3 | 2 | * ----...... | A063524
%Y A366766 16 |!3 | 2 | X---...... | A000105
%Y A366766 17 | 3 | 2 | -X--...... | A365654
%Y A366766 18 | 3 | 2 | * XX--...... | A075679
%Y A366766 19 |!3 | 2 | --X-...... | A000105
%Y A366766 20 |!3 | 2 | X-X-...... | A030222
%Y A366766 21 | 3 | 2 | -XX-...... | A365995
%Y A366766 22 | 3 | 2 | XXX-...... | A365997
%Y A366766 23 | 3 | 2 | ---X...... | A365999
%Y A366766 24 | 3 | 2 | X--X...... | A366001
%Y A366766 25 | 3 | 2 | -X-X...... | A366003
%Y A366766 26 | 3 | 2 | XX-X...... | A366005
%Y A366766 27 | 3 | 2 | * --XX...... | A365652
%Y A366766 28 | 3 | 2 | X-XX...... | A366007
%Y A366766 29 | 3 | 2 | -XXX...... | A366009
%Y A366766 30 | 3 | 2 | * XXXX...... | A365650
%Y A366766 31 | 3 | 3 | * -.-..-.... | A063524
%Y A366766 32 | 3 | 3 | * X.-..-.... | A038119
%Y A366766 33 | 3 | 3 | * -.X..-.... | A038173
%Y A366766 34 | 3 | 3 | * X.X..-.... | A268666
%Y A366766 35 | 3 | 3 | * -.-..X.... | A038171
%Y A366766 36 | 3 | 3 | * X.-..X.... | A363205
%Y A366766 37 | 3 | 3 | * -.X..X.... | A363206
%Y A366766 38 | 3 | 3 | * X.X..X.... | A272368
%Y A366766 39 |!4 | 1 | * --........ | A063524
%Y A366766 40 |!4 | 1 | X-........ | A000012
%Y A366766 41 | 4 | 1 | -X........ | A366340
%Y A366766 42 | 4 | 1 | * XX........ | A365561
%Y A366766 43 |!4 | 2 | * -----..... | A063524
%Y A366766 44 |!4 | 2 | X----..... | A000105
%Y A366766 45 | 4 | 2 | -X---..... | A366338
%Y A366766 46 | 4 | 2 | * XX---..... | A366334
%Y A366766 47 |!4 | 2 | --X--..... | A000105
%Y A366766 48 |!4 | 2 | X-X--..... | A030222
%Y A366766 ...
%Y A366766 75 |!4 | 3 | * ----.--... | A063524
%Y A366766 76 |!4 | 3 | X---.--... | A038119
%Y A366766 77 | 4 | 3 | -X--.--... | A366340
%Y A366766 78 | 4 | 3 | * XX--.--... | A366336
%Y A366766 ...
%Y A366766 139 | 4 | 4 | * -.-..-...- | A063524
%Y A366766 140 | 4 | 4 | * X.-..-...- | A068870
%Y A366766 141 | 4 | 4 | * -.X..-...- | A365356
%Y A366766 142 | 4 | 4 | * X.X..-...- | A365363
%Y A366766 143 | 4 | 4 | * -.-..X...- | A365354
%Y A366766 144 | 4 | 4 | * X.-..X...- | A365361
%Y A366766 145 | 4 | 4 | * -.X..X...- | A365358
%Y A366766 146 | 4 | 4 | * X.X..X...- | A365365
%Y A366766 147 | 4 | 4 | * -.-..-...X | A365353
%Y A366766 148 | 4 | 4 | * X.-..-...X | A365360
%Y A366766 149 | 4 | 4 | * -.X..-...X | A365357
%Y A366766 150 | 4 | 4 | * X.X..-...X | A365364
%Y A366766 151 | 4 | 4 | * -.-..X...X | A365355
%Y A366766 152 | 4 | 4 | * X.-..X...X | A365362
%Y A366766 153 | 4 | 4 | * -.X..X...X | A365359
%Y A366766 154 | 4 | 4 | * X.X..X...X | A365366
%Y A366766 155 |!5 | 1 | * --........ | A063524
%Y A366766 156 |!5 | 1 | X-........ | A000012
%Y A366766 157 | 5 | 1 | -X........ |
%Y A366766 158 | 5 | 1 | * XX........ | A365563
%K A366766 nonn,tabl
%O A366766 1,26
%A A366766 _Pontus von Brömssen_, Oct 22 2023