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
Examples
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.
Links
- 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
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
Comments