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 A380286 #41 Mar 03 2025 10:43:33
%S A380286 1,1,2,3,5,5,5,6,7,7,7,8,9,9,9,10,11,11,11,12,13,13,13,14,15,15,15,16,
%T A380286 17,17,17,18,19,19,19,20,21,21,21,22,23,23,23,24,25,25,25,26,27,27,27,
%U A380286 28,29,29,29,30,31,31,31,32,33,33,33,34,35,35,35,36,37,37,37
%N A380286 Number of distinct values of the number of regions between the free polyominoes with n cells and their bounding boxes.
%C A380286 The regions include any holes in the polyominoes.
%C A380286 From _Andrew Howroyd_, Mar 01 2025: (Start)
%C A380286 Consider the following sequence of polyominoes for n >= 5:
%C A380286 O O O O O O O O O
%C A380286 O O O O O O O O O O O O O O O O O O O O O O O O O O O O
%C A380286 O O O O O O O O
%C A380286 This construction shows how the number of regions between the polyomino and its bounding box can be increased by 2 with the addition of 4 cells. It is also easy to see that any number of fewer holes can also be realized. Moreover, this construction gives the greatest number of regions since except for four corner regions every other region must be bounded on 3 sides by at least one cell separating it from a neighboring region. This leads to a formula for a(n). (End)
%H A380286 Paolo Xausa, <a href="/A380286/b380286.txt">Table of n, a(n) for n = 1..10000</a>
%H A380286 <a href="/index/Pol#polyominoes">Index entries for sequences related to polyominoes</a>.
%H A380286 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).
%F A380286 From _Andrew Howroyd_, Mar 01 2025: (Start)
%F A380286 a(n) = A004525(n + 4) for n >= 5.
%F A380286 G.f.: x*(1 - x + 2*x^2 - x^3 + 2*x^4 - 2*x^5 + x^6 - x^7)/((1 - x)^2*(1 + x^2)). (End)
%F A380286 E.g.f.: (exp(x)*(4 + x) + sin(x))/2 - 2 - 2*x - x^2 - x^3/6 - x^4/24. - _Stefano Spezia_, Mar 03 2025
%e A380286 Illustration for n = 4:
%e A380286 The free polyominoes with four cells are also called free tetrominoes.
%e A380286 The five free tetrominoes are as shown below:
%e A380286 _
%e A380286 |_| _ _ _
%e A380286 |_| |_| |_|_ |_|_ _ _
%e A380286 |_| |_|_ |_|_| |_|_| |_|_|
%e A380286 |_| |_|_| |_| |_| |_|_|
%e A380286 .
%e A380286 The bounding boxes are respectively as shown below:
%e A380286 _
%e A380286 | | _ _ _ _ _ _
%e A380286 | | | | | | | | _ _
%e A380286 | | | | | | | | | |
%e A380286 |_| |_ _| |_ _| |_ _| |_ _|
%e A380286 .
%e A380286 From left to right the number of regions between the free tetrominoes and their bounding boxes are respectively [0, 1, 2, 2, 0], hence there are three distinct values of the number of regions, they are [0, 1, 2], so a(4) = 3.
%e A380286 .
%t A380286 LinearRecurrence[{2, -2, 2, -1}, {1, 1, 2, 3, 5, 5, 5, 6}, 100] (* _Paolo Xausa_, Mar 02 2025 *)
%Y A380286 Row lengths of A380282.
%Y A380286 Cf. A000105, A379627, A379628, A380283, A380284, A380285.
%K A380286 nonn,easy
%O A380286 1,3
%A A380286 _Omar E. Pol_, Jan 24 2025
%E A380286 a(8)-a(16) from _Pontus von Brömssen_, Jan 24 2025
%E A380286 a(17)-a(18) from _John Mason_, Feb 14 2025
%E A380286 a(19) onwards from _Andrew Howroyd_, Feb 17 2025