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 A255748 #41 Nov 03 2022 08:43:20
%S A255748 1,3,4,8,11,13,14,22,29,35,40,44,47,49,50,66,81,95,108,120,131,141,
%T A255748 150,158,165,171,176,180,183,185,186,218,249,279,308,336,363,389,414,
%U A255748 438,461,483,504,524,543,561,578,594,609,623,636,648,659,669,678,686,693,699,704,708,711,713,714,778,841,903,964,1024
%N A255748 Total number of ON states after n generations of cellular automaton based on triangles in a 60-degree wedge (see Comments lines for definition).
%C A255748 Also partial sums of A080079.
%C A255748 In order to construct the structure we use the following rules:
%C A255748 On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.
%C A255748 The nearest triangular cell to the vertex remains OFF.
%C A255748 At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.
%C A255748 At stage n, in the n-th level of the structure, we turn ON k cells connected by their vertices with their bases up, where k = A080079(n).
%C A255748 The cells turned ON remain ON forever.
%C A255748 The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).
%C A255748 Note that this is also the structure in every one of the six wedges of the structure of A256266.
%C A255748 A080079 gives the number of cells turned ON at n-th stage.
%H A255748 Michael De Vlieger, <a href="/A255748/b255748.txt">Table of n, a(n) for n = 1..16384</a>
%H A255748 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 37.
%H A255748 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A255748 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>
%F A255748 a(n) = A256266(n)/6.
%e A255748 Illustration of initial terms:
%e A255748 -----------------------------------------------------------
%e A255748 n A080079(n) a(n) Diagram
%e A255748 -----------------------------------------------------------
%e A255748 . / \
%e A255748 1 1 1 / T \
%e A255748 2 2 3 / T T \
%e A255748 3 1 4 / T \
%e A255748 4 4 8 / T T T T \
%e A255748 5 3 11 / T T T \
%e A255748 6 2 13 / T T \
%e A255748 7 1 14 / T \
%e A255748 8 8 22 / T T T T T T T T \
%e A255748 9 7 29 / T T T T T T T \
%e A255748 10 6 35 / T T T T T T \
%e A255748 11 5 40 / T T T T T \
%e A255748 12 4 44 / T T T T \
%e A255748 13 3 47 / T T T \
%e A255748 14 2 49 / T T \
%e A255748 15 1 50 / T \
%e A255748 ...
%e A255748 For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.
%t A255748 Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 6}] (* _Michael De Vlieger_, Nov 03 2022 *)
%Y A255748 Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.
%K A255748 nonn,look
%O A255748 1,2
%A A255748 _Omar E. Pol_, Mar 30 2015