

A269815


Number of active (ON, black) cells at stage 2^n1 of the twodimensional cellular automaton defined by "Rule 35", based on the 5celled von Neumann neighborhood.


0



1, 5, 37, 185, 817, 3425, 14017, 56705, 228097, 914945, 3664897, 14669825, 58699777, 234840065, 939442177, 3757932545
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OFFSET

0,2


COMMENTS

Initialized with a single black (ON) cell at stage zero.
Lars Blomberg conjectured that Rules 43 and 59 also produce this sequence. It would be nice to have a proof.


REFERENCES

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


LINKS



FORMULA

Conjecture: a(n) = 14*4^(n1)  5*2^n + 1, n>0.  Lars Blomberg, Apr 18 2016
a(n) = 7*a(n1)14*a(n2)+8*a(n3) for n>3.
G.f.: (12*x+16*x^212*x^3) / ((1x)*(12*x)*(14*x)).
(End)


MATHEMATICA

rule=35; 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



EXTENSIONS



STATUS

approved



