A275667 Number of ON cells after n generations in a 2-dimensional "Odd-Rule" cellular automaton on triangular tiling.

1, 3, 7, 9, 7, 21, 25, 27, 7, 21, 49, 63, 25, 75, 103, 81, 7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243, 7, 21, 49, 63, 49, 147, 175, 189, 49, 147, 343, 441, 175, 525, 721, 567, 25, 75, 175, 225, 175, 525, 625, 675, 103, 309, 721

Offset: 0

Author

Kovba Alexey, Aug 04 2016

Keywords

• nonn
• tabf

Comments

Each triangular tile has 3 neighbors. A cell is ON in a given generation if and only if there was an odd number of ON cells among the three nearest neighbors in the preceding generation.

At the initial moment there is a single ON cell.

Given pattern replicates after a number of generations which is a power of 2 when a(n) = 7.

Number of cells on each even step minus one is divisible by 6.

By analogy with the Ekhad, Sloane, Zeilberger link, one may suppose that using ternary expansion of n, recurrence relations for a(n) can be obtained and proved.

From Andrey Zabolotskiy, Aug 04 2016: (Start)

If the first conjecture from the Formula section is true then the fact that the right border of the triangle (see Example) gives A000244 follows directly from it.

If the second conjecture is true then the numbers just before the right border give A102900.

Since the 7 cells which are ON at the beginning of every row are farther and farther away from each other, the n-th term of a row (with offset 0) is a(n)*7 for not very large n.

See also comments to A247666.

(End)

This is ETA rule 170. See the Sadat-Benedek reference for proof of pattern replication. - Paul Cousin, Apr 22 2025

Examples

			From _Omar E. Pol_, Aug 04 2016: (Start)
Written as an irregular triangle in which the row lengths are the terms of A011782 the sequence begins:
1;
3;
7, 9;
7, 21, 25, 27;
7, 21, 49, 63, 25, 75, 103, 81;
7, 21, 49, 63, 49, 147, 175, 189, 25, 75, 175, 225, 103, 309, 409, 243;
...
It appears that the right border gives A000244.
(End)
		

Links

• Paul Cousin, Table of n, a(n) for n = 0..16384
• Paul Cousin, Illustration for n = 0..128
• Paul Cousin, Elementary Triangular Automaton Rule 170
• Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015.
• Alexey Kovba, Illustration for n = 0..5
• MohammadReza Saadat and Benedek Nagy, Copy Machines - Self-reproduction with 2 States on Archimedean Tilings, Journal of Cellular Automata, vol. 17, pp. 221-249, 2023.
• Index to sequences in the OEIS related to cellular automata

Crossrefs

Cf. A160239 (square tiling analog), A247640, A247666 (hexagonal tiling analogs).

Cf. A000244, A011782, A102900, A382971.

Pattern replicating ETA rules: A383369 (rule 90).

Formula

a(0) = 1. Conjecture: a(2*t+1) = 3*a(t).

Conjectures: a(8*t+6) = 3*a(4*t+2) + 4*a(2*t), a(8*t+2) = 3*a(4*t) + 4*a(2*t), a(4*t) = a(2*t). These conjectured formulas together give recurrent relations for a(n) for any n. Also, obviously a(2*n) = A247666(n). - Andrey Zabolotskiy, Aug 04 2016