A169699 - OEIS

A169699

Total number of ON cells at stage n of two-dimensional 5-neighbor outer totalistic cellular automaton defined by "Rule 510".

1, 5, 12, 25, 28, 56, 56, 113, 60, 120, 120, 240, 120, 240, 240, 481, 124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985, 252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

We work on the square grid. Each cell has 4 neighbors, N, S, E, W. If none of your 4 neighbors are ON, your state does not change. If all 4 of your neighbors are ON, your state flips. In all other cases you turn ON. We start with one ON cell.

As observed by Packard and Wolfram (see Fig. 2), a slice along the E-W line shows the successive states of the 1-D CA Rule 126 (see A071035, A071051).

REFERENCES

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

LINKS

Robert Price, Table of n, a(n) for n = 0..128

N. H. Packard and S. Wolfram, Two-Dimensional Cellular Automata, Journal of Statistical Physics, 38 (1985), 901-946.

N. J. A. Sloane, Illustration of first 28 generations

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

Eric Weisstein's World of Mathematics, Elementary Cellular Automaton

S. Wolfram, A New Kind of Science

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

FORMULA

For n>0, it is easy to show that if 2^k <= n < 2^(k+1) then a(n) =

(2^(k+1)-1)*2^(1+wt(n)), where wt is the binary weight A000120, except that if n is a power of 2 we must add 1 to the result.

EXAMPLE

When arranged into blocks of sizes 1,1,2,4,8,16,...:

12, 25,

28, 56, 56, 113,

60, 120, 120, 240, 120, 240, 240, 481,

124, 248, 248, 496, 248, 496, 496, 992, 248, 496, 496, 992, 496, 992, 992, 1985,

252, 504, 504, 1008, 504, 1008, 1008, 2016, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032, 504, 1008, 1008, 2016, 1008, 2016, 2016, 4032,

..., the initial terms in the rows (after the initial rows) have the form 2^m-4 and the final terms are given by A092440. The row beginning with 2^m-4 is divisible by 2^(m-2)-1 (see formula).

MAPLE

A000120 := proc(n) add(i, i=convert(n, base, 2)) end:

ht:=n->floor(log[2](n));

f:=proc(n) local a, t1;

if n=0 then 1 else

a:=(2^(ht(n)+1)-1)*2^(1+A000120(n));

if 2^log[2](n)=n then a:=a+1; fi; a; fi; end;

[seq(f(n), n=0..65)]; # A169699

MATHEMATICA

Map[Function[Apply[Plus, Flatten[ #1]]], CellularAutomaton[{ 510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 100]]

ArrayPlot /@ CellularAutomaton[{510, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, 28]

CROSSREFS