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A131487 a(n) is the number of polyominoes with n edges, including inner edges.

%I A131487 #40 Apr 28 2023 14:50:46
%S A131487 0,0,0,1,0,0,1,0,0,2,0,1,4,0,1,11,1,7,27,4,21,85,21,92,264,89,345,914,
%T A131487 394,1405,3155,1736,5530,11400,7586,22022,41756,32702,87158,156412,
%U A131487 139253,346836,592661,589101,1379837,2275935,2476770,5501846,8830267,10363627,21970992,34594887,43188260,87950618
%N A131487 a(n) is the number of polyominoes with n edges, including inner edges.
%C A131487 An n-celled polyomino with perimeter p has (4n+p)/2 edges. The maximum number of edges in an n-celled polyomino is 3n+1.
%H A131487 Andrew Clarke, <a href="http://www.recmath.com/PolyPages/PolyPages/Isopolyos.html">Isoperimetrical Polyominoes</a>
%F A131487 See A342243 for formula.
%e A131487 A single cell has 4 edges; a domino has 7 edges (this includes the edge between the two cells); both trominoes have 10 edges; their possible orientations are not considered distinct. Thus a(4) = a(7) = 1, a(10) = 2, and a(n) = 0 for n < 10 not equal to 4 or 7.
%e A131487 a(22) = 85 = 83 + 2: there are 83 polyominoes with 7 cells and perimeter 16 (such as a 1 X 7 strip) and two polyominoes with 8 cells and perimeter 12 (a 3 X 3 square without a corner and a 4 X 2 rectangle), and each of these polyominoes has 22 edges.
%e A131487 a(23) = 21. a(24) = 91+1. a(25) = 255+9. a(26) = 89. a(27) = 339+6. a(28) = 847+67. a(34) = 9734+1655+11. a(35) = 7412+174. - _R. J. Mathar_, Feb 22 2021
%Y A131487 Cf. A000105, A057730, A342243.
%Y A131487 Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2), A131488 (analog for hexagonal tiling).
%K A131487 hard,nonn
%O A131487 1,10
%A A131487 _Tanya Khovanova_, Jul 28 2007
%E A131487 a(23)-a(35) from _R. J. Mathar_, Feb 22 2021
%E A131487 a(36)-a(39) from _R. J. Mathar_, Mar 11 2021
%E A131487 a(40)-a(44) from _R. J. Mathar_, Mar 24 2021
%E A131487 a(45)-a(54) from _John Mason_, Apr 28 2023