This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A204804 #30 May 22 2025 04:34:31 %S A204804 1,1,2,4,10,21,49,104,227,468,976,1978,4030,8095,16313,32656,65503, %T A204804 130986,262252,524330,1049054,2097549,4195633,8389840 %N A204804 Number of free tree-like convex polyominoes with n cells. %C A204804 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. %H A204804 Joseph O'Rourke, <a href="http://mathoverflow.net/questions/71032/">MathOverflow Question: Counting restricted polyominoes</a>, July 2011. %F A204804 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 %e A204804 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. %Y A204804 Cf. A131482, A359661. %K A204804 nonn,hard,more %O A204804 1,3 %A A204804 _Joseph O'Rourke_, Jan 19 2012 %E A204804 a(9)-a(16) from _Karl Fabian_, Jan 22 2012 %E A204804 a(17)-a(18) from _John Mason_, May 06 2021 %E A204804 a(19)-a(24) from _Karl Fabian_, May 21 2025