cp's OEIS Frontend

A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.

%I A346800 #26 Apr 15 2023 15:06:03
%S A346800 1,0,2,1,5,4,16,13,54,46,186,167,660,612,2384,2267,8726,8464,32278,
%T A346800 31822,120419,120338,452420,457320,1709845,1745438,6494848,6686929,
%U A346800 24779026,25703792,94899470,99096382,364680344
%N A346800 Number of fixed polyominoes with n cells that have a diagonal axis of symmetry going from lower left to upper right.
%C A346800 This is one of three sequences needed to calculate the number of achiral polyominoes, A030227.  The three sequences together contain exactly two copies of each achiral polyomino. This is the DL sequence in the Shirakawa link. The sequence can be calculated using Redelmeier's method; one chooses an original cell such that no cells in its LL-UR diagonal on one side of it are eligible, nor are any cells in lower LL-UR diagonals. Cells in that original diagonal are counted as one; all others count as two. Jensen's transfer matrix method (see Knuth POLYNUM program) could likely be modified to enumerate this sequence for many more terms; instead of rows, one uses diagonals.
%C A346800 The sequence also enumerates free polyominoes of size 4*n with maximal symmetry that have a center of rotation on a vertex of the underlying square matrix, which are a subset of those enumerated by A142886. - _John Mason_ Jan 27 2022
%H A346800 John Mason, <a href="/A346800/b346800.txt">Table of n, a(n) for n = 1..50</a> (terms 1..47 from Robert A. Russell).
%H A346800 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/programs.html#polyominoes">Program</a>
%H A346800 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 A346800 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.
%F A346800 a(n) = 2*A006748(n) + 2*A056878(n) + A142886(n). - _John Mason_ Jan 27 2022
%e A346800 For a(5)=5, the polyominoes are:  XXX   X     X     XX     X
%e A346800                                     X   X     XX     XX   XXX
%e A346800                                     X   XXX    XX     X    X
%Y A346800 Cf. A030227, A346799, A006748, A056878, A142886.
%K A346800 nonn
%O A346800 1,3
%A A346800 _Robert A. Russell_, Aug 04 2021