A269815 Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 35", based on the 5-celled von Neumann neighborhood.

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

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

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

Conjecture: a(n) = 14*4^(n-1) - 5*2^n + 1, n>0. - Lars Blomberg, Apr 18 2016

Conjectures from Colin Barker, Apr 18 2016: (Start)

a(n) = 7*a(n-1)-14*a(n-2)+8*a(n-3) for n>3.

G.f.: (1-2*x+16*x^2-12*x^3) / ((1-x)*(1-2*x)*(1-4*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

Cf. A269814.

Sequence in context: A270898 A273251 A270326 * A270931 A241629 A273539

Adjacent sequences: A269812 A269813 A269814 * A269816 A269817 A269818

Keyword

nonn,more

Author

Robert Price, Mar 05 2016

Extensions

Corrected a(8) and a(9)-a(15) from Lars Blomberg, Apr 18 2016

Status

approved