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 A380285 #27 Feb 14 2025 10:47:29
%S A380285 0,0,1,5,21,71,255,961,3630,13973,53938,209641,815784,3183642,
%T A380285 12439291,48686549,190787588,748645732
%N A380285 Total number of regions between the free polyominoes with n cells and their bounding boxes.
%C A380285 The regions include any holes in the polyominoes.
%H A380285 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%F A380285 a(n) = Sum_{k>0} k*A380282(n,k). - _Pontus von Brömssen_, Jan 24 2025
%e A380285 Illustration for n = 4:
%e A380285 The free polyominoes with four cells are also called free tetrominoes.
%e A380285 The five free tetrominoes are as shown below:
%e A380285 _
%e A380285 |_| _ _ _
%e A380285 |_| |_| |_|_ |_|_ _ _
%e A380285 |_| |_|_ |_|_| |_|_| |_|_|
%e A380285 |_| |_|_| |_| |_| |_|_|
%e A380285 .
%e A380285 The bounding boxes are respectively as shown below:
%e A380285 _
%e A380285 | | _ _ _ _ _ _
%e A380285 | | | | | | | | _ _
%e A380285 | | | | | | | | | |
%e A380285 |_| |_ _| |_ _| |_ _| |_ _|
%e A380285 .
%e A380285 From left to right the number of regions between the free tetrominoes and their bounding boxes are respectively 0, 1, 2, 2, 0. Hence the total number of regions is 0 + 1 + 2 + 2 + 0 = 5, so a(4) = 5.
%e A380285 .
%Y A380285 Row sums of A380283 and of A380284.
%Y A380285 Cf. A379628 (total area of the regions).
%Y A380285 Cf. A000105, A057766, A379623, A379624, A379625, A379626, A379627, A379629, A379637, A379638, A380282.
%K A380285 nonn,more
%O A380285 1,4
%A A380285 _Omar E. Pol_, Jan 18 2025
%E A380285 a(8)-a(16) from _Pontus von Brömssen_, Jan 24 2025
%E A380285 a(17)-a(18) from _John Mason_, Feb 14 2025