

A269708


Number of active (ON,black) cells at stage 2^n1 of the twodimensional cellular automaton defined by "Rule 14", based on the 5celled von Neumann neighborhood.


0



1, 5, 20, 76, 292, 1132, 4420, 17356, 68452, 270892, 1074820, 4273036, 17013412, 67817452, 270561220, 1080119116
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

Initialized with a single black (ON) cell at stage zero.
Rules 46, 142 and 174 also generate this sequence.


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 5Neighbor Cellular Automata
Index to Elementary Cellular Automata


FORMULA

Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 4*3^(n2)+4^n for n>1.
a(n) = 7*a(n1)12*a(n2) for n>3.
G.f.: (12*x3*x^24*x^3) / ((13*x)*(14*x)).
(End)


MATHEMATICA

rule=14; 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. A269707.
Sequence in context: A005283 A057552 A300918 * A295347 A270985 A289786
Adjacent sequences: A269705 A269706 A269707 * A269709 A269710 A269711


KEYWORD

nonn,more


AUTHOR

Robert Price, Mar 04 2016


EXTENSIONS

a(9)a(15) from Lars Blomberg, Apr 12 2016


STATUS

approved



