A255748 - OEIS

%I #41 Nov 03 2022 08:43:20

%S 1,3,4,8,11,13,14,22,29,35,40,44,47,49,50,66,81,95,108,120,131,141,

%T 150,158,165,171,176,180,183,185,186,218,249,279,308,336,363,389,414,

%U 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 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 Also partial sums of A080079.

%C In order to construct the structure we use the following rules:

%C On the infinite triangular grid we are in a 60-degree wedge with the vertex located on top of the wedge.

%C The nearest triangular cell to the vertex remains OFF.

%C At stage 1, we turn ON the cell whose base is adjacent to the previous OFF cell.

%C 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 The cells turned ON remain ON forever.

%C The structure seems to grow into the holes of a virtual Sierpiński's triangle (see example).

%C Note that this is also the structure in every one of the six wedges of the structure of A256266.

%C A080079 gives the number of cells turned ON at n-th stage.

%H Michael De Vlieger, <a href="/A255748/b255748.txt">Table of n, a(n) for n = 1..16384</a>

%H 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 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 <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%F a(n) = A256266(n)/6.

%e Illustration of initial terms:

%e -----------------------------------------------------------

%e n   A080079(n)   a(n)                  Diagram

%e -----------------------------------------------------------

%e .                                        / \

%e 1       1         1                     / T \

%e 2       2         3                    / T T \

%e 3       1         4                   /   T   \

%e 4       4         8                  / T T T T \

%e 5       3        11                 /   T T T   \

%e 6       2        13                /     T T     \

%e 7       1        14               /       T       \

%e 8       8        22              / T T T T T T T T \

%e 9       7        29             /   T T T T T T T   \

%e 10      6        35            /     T T T T T T     \

%e 11      5        40           /       T T T T T       \

%e 12      4        44          /         T T T T         \

%e 13      3        47         /           T T T           \

%e 14      2        49        /             T T             \

%e 15      1        50       /               T               \

%e ...

%e For n = 15 after 15 generations there are 50 ON cells in the structure, so a(15) = 50.

%t Accumulate@ Flatten@ Table[Range[2^n, 1, -1], {n, 0, 6}] (* _Michael De Vlieger_, Nov 03 2022 *)

%Y Cf. A047999, A001316, A080079, A139250, A169779, A169788, A170905, A233970, A256256, A256266.

%K nonn,look

%O 1,2

%A _Omar E. Pol_, Mar 30 2015