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A273314 Partial sums of the number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood. 1
1, 6, 23, 64, 137, 250, 411, 628, 909, 1262, 1695, 2216, 2833, 3554, 4387, 5340, 6421, 7638, 8999, 10512, 12185, 14026, 16043, 18244, 20637, 23230, 26031, 29048, 32289, 35762, 39475, 43436, 47653, 52134, 56887, 61920, 67241, 72858, 78779, 85012, 91565, 98446 (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

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, May 19 2016: (Start)

a(n) = (4*n^3+12*n^2-13*n+15)/3 for n>0.

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4) for n>4.

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

(End)

MATHEMATICA

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

code=643; 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}];

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

Table[Total[Part[on, Range[1, i]]], {i, 1, Length[on]}] (* Sum at each stage *)

CROSSREFS

Cf. A166147.

Sequence in context: A273252 A208598 A119712 * A281424 A005745 A213557

Adjacent sequences:  A273311 A273312 A273313 * A273315 A273316 A273317

KEYWORD

nonn,easy

AUTHOR

Robert Price, May 19 2016

STATUS

approved

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Last modified December 12 06:05 EST 2017. Contains 295937 sequences.