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

0,2

Comments

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.

Links

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

Formula

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

Example

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.

Mathematica

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 *)

Prog

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

Crossrefs

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

Sequence in context: A017593 A335908 A096286 * A344596 A280802 A124353

Adjacent sequences: A256253 A256254 A256255 * A256257 A256258 A256259

Keyword

nonn

Author

Omar E. Pol, Mar 20 2015

Status

approved