A204804 Number of free tree-like convex polyominoes with n cells.

1, 1, 2, 4, 10, 21, 49, 104, 227, 468, 976, 1978, 4030, 8095, 16313, 32656, 65503, 130986, 262252, 524330, 1049054, 2097549, 4195633, 8389840

Offset

1,3

Comments

Free: none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. Tree-like: never does a 2x2 subarrangement of squares occur in the shape. So the dual graph is a tree. Convex: every horizontal, or vertical line, meets the shape in either a single segment, or not at all.

Links

Table of n, a(n) for n=1..24.

Joseph O'Rourke, MathOverflow Question: Counting restricted polyominoes, July 2011.

Formula

It seems that a(n) = 2^(n-1) + 2^(ceiling(n/2)-1) - b(n), where the g.f. of b(n) is x*(1+x^5+x^6) / ((1-x)^4*(1+x)^2*(1+x^2)), and accordingly this sequence itself is a linear recurrence of order 11 with signature (4,-2,-10,14,-2,-8,14,-13,-2,10,-4); cf. Gerhard Paseman's answer at MathOverflow. - Andrei Zabolotskii, May 21 2025

Example

n=1: one square. n=2: a 2x1 rectangle. n=3: a 3x1 rectangle; an L-shape. So the sequence starts: 1,1,2,... Images up to n=8 at the MathOverflow link.

Crossrefs

Cf. A131482, A359661.

Sequence in context: A018003 A328692 A255711 * A328693 A165136 A165137

Adjacent sequences: A204801 A204802 A204803 * A204805 A204806 A204807

Keyword

nonn,hard,more

Author

Joseph O'Rourke, Jan 19 2012

Extensions

a(9)-a(16) from Karl Fabian, Jan 22 2012

a(17)-a(18) from John Mason, May 06 2021

a(19)-a(24) from Karl Fabian, May 21 2025

Status

approved