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 A343208 #22 Apr 10 2021 02:03:17
%S A343208 1,5,27,143,744,3832,19636,100348,511969,2608905,13282011,67567527,
%T A343208 343510966,1745495390,8865633276,45013599940,228478238613,
%U A343208 1159398424925,5881978415019,29835289653043,151308803657699,767245632538063,3889991549017581,19720295705928713,99961847384995974
%N A343208 a(n) = Sum_{k=1..n} k*A001168(k)*binomial(n-1,k-1), where A001168(k) is the number of fixed polyominoes with k cells.
%C A343208 This is the number of ways n blocks can be placed on a 2D grid such that, after the first block, each block touches at least one face of a previously placed block, and each block either touches the ground plane or is supported by a block below it. See the attached file for a derivation.
%C A343208 The number of ways n squares can be placed similarly on a 1D line is given by A001792.
%H A343208 Scott R. Shannon, <a href="/A343208/a343208_1.txt">Derivation of the title formula</a>.
%e A343208 Considering the sequence as face-touching blocks:
%e A343208 a(1) = 1 as a single block can be placed in one way.
%e A343208 a(2) = 5 as, after the first block is placed, the second block can be placed so that it touches the ground plane and one of the four sides of the first block, or it can be placed directly on top of the first block, giving five total arrangements.
%e A343208 a(3) = 27 as the third block can be placed in one way directly on top of the tower of the two previous blocks, on the ground next to the tower of two blocks in four ways, next to one of the three faces of the second block on the ground plane or on top of the second block in 4*4 = 16 total ways, or on the ground plane touching one of the faces of the first block with the second block touching one of the other faces of the first block in 6 total ways. Summing the configurations gives 27 total ways the three blocks can be arranged.
%Y A343208 Cf. A001168, A001792, A007318.
%K A343208 nonn
%O A343208 1,2
%A A343208 _Scott R. Shannon_ and _Zach J. Shannon_, Apr 08 2021