cp's OEIS Frontend

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).

%I A006747 M3741 #75 Oct 19 2023 12:10:24
%S A006747 0,0,0,1,1,5,4,18,19,73,73,278,283,1076,1090,4125,4183,15939,16105,
%T A006747 61628,62170,239388,240907,932230,936447,3641945,3651618,14262540,
%U A006747 14277519,55987858,55961118,220223982,219813564,867835023,865091976,3425442681
%N 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).
%C A006747 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.
%C A006747 The rotation center of a polyomino with this symmetry may lie at 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, reformulated by _Günter Rote_, Oct 19 2023
%D A006747 S. W. Golomb, Polyominoes, Princeton Univ. Press, NJ, 1994.
%D A006747 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006747 John Mason, <a href="/A006747/b006747.txt">Table of n, a(n) for n = 1..50</a>
%H A006747 Tomás Oliveira e Silva, <a href="http://sweet.ua.pt/tos/animals.html">Enumeration of polyominoes</a>
%H A006747 Tomás Oliveira e Silva, <a href="/A006747/a006747.png">Numbers of polyominoes classified according to Redelmeier's symmetry classes</a> (an extract from the previous link)
%H A006747 D. H. Redelmeier, <a href="http://dx.doi.org/10.1016/0012-365X(81)90237-5">Counting polyominoes: yet another attack</a>, Discrete Math., 36 (1981), 191-203.
%H A006747 D. H. Redelmeier, <a href="/A056877/a056877.png">Table 3</a> of Counting polyominoes...
%F A006747 a(n) = A351615(n) + A234008(n/2) + A351616(n/2) for even n, otherwise a(n) = A351615(n). - _John Mason_, Feb 17 2022
%e A006747 a(2) = 0 because the "domino" polyomino has symmetry group of order 4.
%e A006747 For n=3, the three-celled polyomino [ | | ] has group of order 4, and the polyomino
%e A006747 . [ ]
%e A006747 . [ | ]
%e A006747 has only reflective symmetry, so a(3) = 0.
%e A006747 a(4) = 1 because of (in Golomb's notation) the "skew tetromino".
%Y A006747 Cf. A000105, A001168, A006746, A056877, A006748, A056878, A006747, A006749.
%Y A006747 Sequences classifying polyominoes by symmetry group: A000105, A006746, A006747, A006748, A006749, A056877, A056878, A142886, A144553, A144554, A351615, A234008, A351616.
%Y A006747 Polyomino rings of length 2n with twofold rotational symmetry: A348402, A348403, A348404.
%K A006747 nonn
%O A006747 1,6
%A A006747 _N. J. A. Sloane_
%E A006747 Extended to n=28 by Tomás Oliveira e Silva
%E A006747 a(1)-a(3) prepended by _Andrew Howroyd_, Dec 04 2018
%E A006747 Edited by _N. J. A. Sloane_, Nov 28 2020
%E A006747 a(29)-a(36) from _John Mason_, Oct 16 2021