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A264797 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 342", based on the 5-celled von Neumann neighborhood. 3
1, 5, 8, 24, 17, 53, 32, 96, 49, 149, 72, 216, 97, 293, 128, 384, 161, 485, 200, 600, 241, 725, 288, 864, 337, 1013, 392, 1176, 449, 1349, 512, 1536, 577, 1733, 648, 1944, 721, 2165, 800, 2400, 881, 2645, 968, 2904, 1057, 3173, 1152, 3456, 1249, 3749, 1352 (list; graph; refs; listen; history; text; internal format)
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

Robert Price, Table of n, a(n) for n = 0..128

Robert Price, Diagrams of the first 20 stages.

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

Conjectures from Colin Barker, Apr 03 2016: (Start)

a(n) = 2*a(n-2)-2*a(n-6)+a(n-8) for n>7.

G.f.: (1+5*x+6*x^2+14*x^3+x^4+5*x^5) / ((1-x)^3*(1+x)^3*(1+x^2)).

(End)

MATHEMATICA

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];

code=342; stages=128;

rule=IntegerDigits[code, 2, 10];

g=2*stages+1; (* Maximum size of grid *)

a=PadLeft[{{1}}, {g, g}, 0, Floor[{g, g}/2]]; (* Initial ON cell on grid *)

ca=a;

ca=Table[ca=CAStep[rule, ca], {n, 1, stages+1}];

PrependTo[ca, a];

(* Trim full grid to reflect growth by one cell at each stage *)

k=(Length[ca[[1]]]+1)/2;

ca=Table[Table[Part[ca[[n]][[j]], Range[k+1-n, k-1+n]], {j, k+1-n, k-1+n}], {n, 1, k}];

Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)

CROSSREFS

Sequence in context: A192651 A105963 A270125 * A270905 A253078 A270185

Adjacent sequences:  A264794 A264795 A264796 * A264798 A264799 A264800

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 03 2016

STATUS

approved

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Last modified July 6 12:26 EDT 2022. Contains 355110 sequences. (Running on oeis4.)