|
|
A006747
|
|
Number of rotationally symmetric polyominoes with n cells (that is, polyominoes with exactly the symmetry group C_2 generated by a 180-degree rotation).
(Formerly M3741)
|
|
27
|
|
|
0, 0, 0, 1, 1, 5, 4, 18, 19, 73, 73, 278, 283, 1076, 1090, 4125, 4183, 15939, 16105, 61628, 62170, 239388, 240907, 932230, 936447, 3641945, 3651618, 14262540, 14277519, 55987858, 55961118, 220223982, 219813564, 867835023, 865091976, 3425442681
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,6
|
|
COMMENTS
|
This sequence gives the number of free polyominoes with symmetry group "R" in Redelmeier's notation. See his Tables 1 and 3, also the column "Rot" in Oliveira e Silva's table.
Polyominoes having this symmetry may have an axis of symmetry that coincides with the center of a square, the middle of an edge, or a vertex of a square. These subsets are enumerated by A351615, A234008 and A351616 respectively. - John Mason, Feb 17 2022
|
|
REFERENCES
|
S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
LINKS
|
D. H. Redelmeier, Table 3 of Counting polyominoes...
|
|
FORMULA
|
|
|
EXAMPLE
|
a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
. [ ]
. [ | ]
has only reflective symmetry, so a(3) = 0.
a(4) = 1 because of (in Golomb's notation) the "skew tetromino".
|
|
CROSSREFS
|
Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Extended to n=28 by Tomás Oliveira e Silva
|
|
STATUS
|
approved
|
|
|
|