A247640 Number of ON cells after n generations of "Odd-Rule" cellular automaton on hexagonal lattice based on 6-celled neighborhood.

1, 6, 6, 24, 6, 36, 24, 96, 6, 36, 36, 144, 24, 144, 96, 384, 6, 36, 36, 144, 36, 216, 144, 576, 24, 144, 144, 576, 96, 576, 384, 1536, 6, 36, 36, 144, 36, 216, 144, 576, 36, 216, 216, 864, 144, 864, 576, 2304, 24, 144, 144, 576, 144, 864

Offset

0,2

Comments

The neighborhood of a cell consists of the six surrounding cells (but not the cell itself). A cell is ON at generation n iff an odd number of its neighbors were ON at the previous generation. We start with one ON cell.

This is the Run Length Transform of the sequence 1, 6, 24, 96, 384, 1536, 6144, 24576, ... (almost certainly A164908, or 1 followed by A002023).

It appears that this is also the sequence corresponding to the odd-rule cellular automaton defined by OddRule 356 (see Ekhad-Sloane-Zeilberger "Odd-Rule Cellular Automata on the Square Grid" link). - N. J. A. Sloane, Feb 26 2015

Links

Table of n, a(n) for n=0..53.

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; see also the Accompanying Maple Package.

Shalosh B. Ekhad, N. J. A. Sloane, and  Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.

N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Index entries for sequences related to cellular automata

Formula

a(n) = number of terms in expansion of f^n mod 2, where f = 1/x+x+1/y+y+1/(x*y)+x*y (mod 2);

Maple

C := f->`if`(type(f, `+`), nops(f), 1);
f := 1/x+x+1/y+y+1/(x*y)+x*y;
g := n->expand(f^n) mod 2;
[seq(C(g(n)), n=0..100)];

Mathematica

A247640[n_] := Total[CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, {{{n}}}], 2]; Array[A247640, 54, 0] (* JungHwan Min, Sep 06 2016 *)
A247640L[n_] := Total[#, 2] & /@ CellularAutomaton[{42, {2, {{1, 1, 0}, {1, 0, 1}, {0, 1, 1}}}, {1, 1}}, {{{1}}, 0}, n]; A247640L[53] (* JungHwan Min, Sep 06 2016 *)

Crossrefs

Cf. A164908, A001023, A071053, A160239, A247666.

Sequence in context: A255295 A255475 A253100 * A255470 A267710 A306896

Adjacent sequences: A247637 A247638 A247639 * A247641 A247642 A247643

Keyword

nonn

Author

N. J. A. Sloane, Sep 22 2014

Status

approved