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%I #6 Dec 04 2021 12:44:01
%S 1,0,0,0,1,0,0,1,2,0,0,3,6,0,0,10,18,0,0,35,57,0,0,126,191,0,0,461,
%T 658,0,0,1699,2308,0,0,6315,8241,0,0,23686,29853,0,0,89432,109268,0,0,
%U 339473,403450,0,0,1294826,1501074
%N Number of fixed polyominoes with n cells that have fourfold rotational symmetry centered at the center of a cell.
%C These are polyominoes of the regular tiling with Schläfli symbol {4,4}. Chiral pairs are counted as two. This is one of the five sequences, along with A001168, needed to calculate the number of oriented polyominoes, A000988. It is the F90 sequence in the Shirakawa link. The calculation follows Redelmeier's method of determining inner rings.
%H Robert A. Russell, <a href="/A348849/b348849.txt">Table of n, a(n) for n = 1..96</a>
%H 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 Toshihiro Shirakawa, <a href="https://www.gathering4gardner.org/g4g10gift/math/Shirakawa_Toshihiro-Harmonic_Magic_Square.pdf">Enumeration of Polyominoes considering the symmetry</a>, April 2012, pp. 3-4.
%e For a(9)=2, the polyomino is a 3 X 3 square or a row and column of five cells sharing their central cells.
%Y Cf. A000988, A144553, A348848 (vertex center).
%Y Inner rings: A324406, A324407, A324408, A324409.
%K nonn
%O 1,9
%A _Robert A. Russell_, Nov 01 2021