A256257 -id:A256257 - cp's OEIS frontend

A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

0, 6, 18, 30, 54, 66, 90, 114, 162, 174, 198, 222, 270, 294, 342, 390, 486, 498, 522, 546, 594, 618, 666, 714, 810, 834, 882, 930, 1026, 1074, 1170, 1266, 1458, 1470, 1494, 1518, 1566, 1590, 1638, 1686, 1782, 1806, 1854, 1902, 1998, 2046, 2142, 2238, 2430, 2454, 2502, 2550, 2646, 2694, 2790, 2886, 3078, 3126, 3222, 3318, 3510, 3606, 3798, 3990, 4374

Offset: 0

Omar E. Pol, Mar 20 2015

nonn

Analog of A160720, but here we are working on the triangular lattice.
The first differences (A256257) gives the number of triangular cells turned ON at every generation.
Also 6 times the sum of all entries in rows 0 to n of Sierpiński's triangle A047999.
Also 6 times the total number of odd entries in first n rows of Pascal's triangle A007318, see formula section.
The structure contains three distinct kinds of polygons formed by triangular ON cells: the initial hexagon, rhombuses (each one formed by two ON cells) and unit triangles.
Note that if n is a power of 2 greater than 2, the structure looks like concentric hexagons with triangular holes, where some of them form concentric stars.

			On the infinite triangular grid we start with all triangular cells turned OFF, so a(0) = 0.
At stage 1, in the structure there are six triangular cells turned ON forming a regular hexagon, so a(1) = 6.
At stage 2, there are 12 new triangular ON cells forming six rhombuses around the initial hexagon, so a(2) = 6 + 12 = 18.
And so on.

Michael De Vlieger, Table of n, a(n) for n = 0..16386
Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 30.
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Index entries for sequences related to cellular automata

Cf. A001316, A006046, A007318, A025192, A047999, A151723, A160120, A160720, A161644, A256257.

Prepend[6*FoldList[Plus, 0, Total /@ CellularAutomaton[90, Join[Table[0, {#}], {1}, Table[0, {#}]], #]][[2 ;; -1]], 0] &[63] (* Michael De Vlieger, Nov 03 2022, after Bradley Klee at A006046 *)

a(n) = 6*sum(j=0, n, sum(k=0, j, binomial(j, k) % 2)); \\ Michel Marcus, Apr 01 2015

a(n) = 6*A006046(n).

A262617 First differences of A256266.

Original entry on oeis.org

0, 6, 12, 6, 24, 18, 12, 6, 48, 42, 36, 30, 24, 18, 12, 6, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 192, 186, 180, 174, 168, 162, 156, 150, 144, 138, 132, 126, 120, 114, 108, 102, 96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6, 384, 378, 372, 366, 360, 354, 348, 342, 336, 330, 324, 318

Offset: 0

Omar E. Pol, Oct 02 2015

Number of cells turned ON at n-th stage of the cellular automaton of A256266.

			With the terms written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
0;
6;
12, 6;
24, 18, 12, 6;
48, 42, 36, 30, 24, 18, 12, 6;
96, 90, 84, 78, 72, 66, 60, 54, 48, 42, 36, 30, 24, 18, 12, 6;
...
Apart from the initial zero the rows list the initial terms of the positive multiples of 6 in decreasing order.

Cf. A008588, A011782, A080079, A255748, A256257, A256266.

a(n) = if(n==0, 0, -6*n+12*2^floor(log(n)/log(2)));
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

a(n) = 6 * A080079(n), n >= 1.

cp's OEIS Frontend

A256256 Total number of ON cells after n generations of cellular automaton on triangular grid, starting from a node, in which every 60-degree wedge looks like the Sierpiński's triangle.

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