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A269708
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Number of active (ON,black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.
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0
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1, 5, 20, 76, 292, 1132, 4420, 17356, 68452, 270892, 1074820, 4273036, 17013412, 67817452, 270561220, 1080119116
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OFFSET
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0,2
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COMMENTS
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Initialized with a single black (ON) cell at stage zero.
Rules 46, 142 and 174 also generate this sequence.
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REFERENCES
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S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
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LINKS
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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
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FORMULA
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Conjectures from Colin Barker, Mar 08 2016: (Start)
a(n) = 4*3^(n-2)+4^n for n>1.
a(n) = 7*a(n-1)-12*a(n-2) for n>3.
G.f.: (1-2*x-3*x^2-4*x^3) / ((1-3*x)*(1-4*x)).
(End)
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A269707.
Sequence in context: A005283 A057552 A300918 * A295347 A270985 A289786
Adjacent sequences: A269705 A269706 A269707 * A269709 A269710 A269711
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KEYWORD
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nonn,more
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AUTHOR
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Robert Price, Mar 04 2016
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EXTENSIONS
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a(9)-a(15) from Lars Blomberg, Apr 12 2016
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STATUS
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approved
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