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A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.

%I A234006 #50 Jul 26 2024 02:31:52
%S A234006 1,2,4,11,35,114,392,1381,4998,18292,67791,253182,952527,3603389,
%T A234006 13699516,52300071,200406183,770424072,2970400815,11482442855,
%U A234006 44491876993,172766491178,672186631950,2619995178793,10228902801505,39996341268584,156612023001490,614044347934591
%N A234006 Free polyominoes with 2n squares, having reflectional symmetry on axis that coincides with edges.
%C A234006 The number of free polyominoes of size 2n that have reflectional symmetry on a horizontal or vertical axis that coincides with the edges of some of the squares. The sequence is defined for 2n rather than n as odd-sized polyominoes cannot have the required symmetry.
%H A234006 John Mason, <a href="/A234006/b234006.txt">Table of n, a(n) for n = 1..50</a>
%F A234006 a(2*n+1) = A151525(2*n+1), a(2*n) = A151525(2*n) + A182645(n) - A001168(n). - _Andrew Howroyd_, Dec 05 2018
%F A234006 If n odd, a(n) = A349329(n) + A346799(n), otherwise a(n) = A349329(n) + A346799(n) + A346800(n/2) + A351191(n/2). - _John Mason_, Mar 15 2023
%t A234006 A151525 = Cases[Import["https://oeis.org/A151525/b151525.txt", "Table"], {_, _}][[All, 2]];
%t A234006 A182645 = Cases[Import["https://oeis.org/A182645/b182645.txt", "Table"], {_, _}][[All, 2]];
%t A234006 A001168 = Cases[Import["https://oeis.org/A001168/b001168.txt", "Table"], {_, _}][[All, 2]];
%t A234006 a[n_] := If[OddQ[n], A151525[[n]], A151525[[n]] + A182645[[n/2]] - A001168[[n/2]]];
%t A234006 Array[a, 28] (* _Jean-François Alcover_, Sep 10 2019, after _Andrew Howroyd_ *)
%Y A234006 Cf. A000105, A001168, A001933, A151525, A182645, A234007, A234008, A234009, A234010, A349329, A346799, A346800, A351191.
%K A234006 nonn
%O A234006 1,2
%A A234006 _John Mason_, Dec 18 2013
%E A234006 a(12)-a(28) from _Andrew Howroyd_, Dec 05 2018