A269811 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 33", based on the 5-celled von Neumann neighborhood.

1, 4, 32, 164, 732, 3084, 12692, 51604, 208492, 839324, 3371652, 13526244, 54217052, 217190764, 869703412, 3481580084

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

References

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

Links

Table of n, a(n) for n=0..15.

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

Conjectures from Colin Barker, Apr 15 2016: (Start)

a(n) = -2+(-1)^n/3-(5*2^(1+n))/3-5*3^(-2+n)+13*4^(-1+n) for n>1.

a(n) = (117*2^(2*n)-15*2^(n+3)-20*3^n-60)/36 for n>1 and even.

a(n) = (117*2^(2*n)-15*2^(n+3)-20*3^n-84)/36 for n>1 and odd.

G.f.: (1-5*x+21*x^2-39*x^3-30*x^4+36*x^5+40*x^6) / ((1-x)*(1+x)*(1-2*x)*(1-3*x)*(1-4*x)).

(End)

Mathematica

rule=33; stages=300;
ca=CellularAutomaton[{rule, {2, {{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}}, {1, 1}}, {{{1}}, 0}, stages]; (* Start with single black cell *)
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Part[on, 2^Range[0, Log[2, stages]]] (* Extract relevant terms *)

Crossrefs

Cf. A269810.

Sequence in context: A336448 A259854 A272424 * A271126 A271804 A270207

Adjacent sequences: A269808 A269809 A269810 * A269812 A269813 A269814

Keyword

nonn,more

Author

Robert Price, Mar 05 2016

Extensions

corrected a(8) and a(9)-a(15) from Lars Blomberg, Apr 15 2016

Status

approved