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
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
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
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