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

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, 394, 1405, 3155, 1736, 5530, 11400, 7586, 22022, 41756, 32702, 87158, 156412, 139253, 346836, 592661, 589101, 1379837, 2275935, 2476770, 5501846, 8830267, 10363627, 21970992, 34594887, 43188260, 87950618

Offset: 1

Author

Tanya Khovanova, Jul 28 2007

Keywords

• hard
• nonn

Comments

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.

Examples

			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.
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.
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
		

Links

• Andrew Clarke, Isoperimetrical Polyominoes

Crossrefs

Cf. A000105, A057730, A342243.

Cf. A131482 (number of n-celled polyominoes with perimeter 2n+2), A131488 (analog for hexagonal tiling).

Formula

See A342243 for formula.

Extensions

a(23)-a(35) from R. J. Mathar, Feb 22 2021

a(36)-a(39) from R. J. Mathar, Mar 11 2021

a(40)-a(44) from R. J. Mathar, Mar 24 2021

a(45)-a(54) from John Mason, Apr 28 2023