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.

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

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