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A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).

%I A121198 #56 Dec 23 2024 14:53:42
%S A121198 2,1,4,10,36,110,392,1371,5000,18251,67792,253040,952540,3602846,
%T A121198 13699554,52298057,200406388,770416390,2970401696,11482413680,
%U A121198 44491881090,172766379334,672186650116,2619994749395,10228902882212,39996339612824,156612023354364,614044341535992
%N A121198 Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).
%C A121198 Consider the tiling of the plane with squares of two different sizes as seen for example in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. Sequence gives the number of "n-PairSquares", that is, polyominoes or animals that can be formed on this tiling from "n big or little squares, where the conjunction between two squares must involve an entire edge at least". - Original description (_N. J. A. Sloane_, Aug 17 2006, with quote from Livio Zucca's site)
%C A121198 Also counts one-sided polyominoes cut from an infinite chessboard with the usual coloring (big and little squares in Fig. 2.4.2(g) of Grünbaum and Shephard are equivalent to the two colors on a chessboard, and ignoring connections that are not a whole edge of one square means the connectivity is also equivalent); see Myers link regarding difference from A001071 for even terms a(6) onwards. - _Joseph Myers_, Oct 01 2011
%D A121198 Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987.
%H A121198 John Mason, <a href="/A121198/b121198.txt">Table of n, a(n) for n = 1..50</a>
%H A121198 Joseph Myers, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2010-November/013893.html">Chessboard polyominoes</a>
%H A121198 Livio Zucca, <a href="http://www.iread.it/lz/polymultiforms2.html">PolyMultiForms</a>
%F A121198 From _John Mason_, Dec 24 2021: (Start)
%F A121198 For odd n, a(n) = 2*A000105(n) + 2*A030228(n).
%F A121198 For n multiple of 2 but not of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2).
%F A121198 For n multiple of 4, a(n) = 2*A000105(n) + 2*A030228(n) - A346799(n/2) - 2*A234008(n/2) - A234009(n/4) - A234007(n/4). (End)
%Y A121198 Cf. A001071, A001933, A121195, A121196, A000105 (free polyominoes), A030228 (chiral polyominoes), A234009 (free polyominoes with 90-degree rotational symmetry about a square corner), A234007 (chiral polyominoes with 90-degree rotational symmetry about a square corner), A346799 (achiral polyominoes with twofold rotational symmetry around the center of an edge), A234008 (chiral polyominoes with 180-degree rotational symmetry about the center of an edge).
%K A121198 nonn,hard
%O A121198 1,1
%A A121198 _N. J. A. Sloane_, Aug 17 2006
%E A121198 a(6)-a(17) by _Joseph Myers_, Oct 01 2011
%E A121198 a(18)-a(21) by _John Mason_, Jan 04 2014
%E A121198 Erroneous a(21) removed by _John Mason_, Feb 12 2021
%E A121198 a(21)-a(28) from _John Mason_, Dec 24 2021