

A273314


Partial sums of the number of active (ON,black) cells in nth stage of growth of twodimensional cellular automaton defined by "Rule 643", based on the 5celled 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
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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 5Neighbor Cellular Automata
Index to Elementary Cellular Automata


FORMULA

Conjectures from Colin Barker, May 19 2016: (Start)
a(n) = (4*n^3+12*n^213*n+15)/3 for n>0.
a(n) = 4*a(n1)6*a(n2)+4*a(n3)a(n4) for n>4.
G.f.: (1+2*x+5*x^2+4*x^34*x^4) / (1x)^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+1n, k1+n]], {j, k+1n, k1+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



