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A271902 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 427", based on the 5-celled von Neumann neighborhood. 4
1, 5, 13, 32, 37, 89, 80, 145, 113, 292, 169, 357, 244, 581, 293, 688, 397, 969, 472, 1105, 585, 1476, 673, 1621, 828, 2069, 909, 2256, 1093, 2761, 1200, 2977, 1393, 3572, 1513, 3797, 1748, 4469, 1861, 4736, 2125, 5465, 2264, 5761, 2537, 6580, 2689, 6885 (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

Empirical g.f.: (1 +6*x +18*x^2 +44*x^3 +61*x^4 +96*x^5 +88*x^6 +68*x^7 +7*x^8 +26*x^9 +6*x^10 -5*x^12) / ((1 -x)^3*(1 +x)^3*(1 +x^2)^2*(1 +x +x^2)). - Colin Barker, Apr 21 2016

MATHEMATICA

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

code=427; 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: A062403 A066688 A046789 * A272539 A066184 A231799

Adjacent sequences:  A271899 A271900 A271901 * A271903 A271904 A271905

KEYWORD

nonn,easy

AUTHOR

Robert Price, Apr 20 2016

STATUS

approved

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Last modified April 20 02:06 EDT 2019. Contains 322291 sequences. (Running on oeis4.)