A272221 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 441", based on the 5-celled von Neumann neighborhood.

1, 4, 9, 36, 25, 96, 65, 192, 97, 284, 149, 436, 181, 580, 305, 812, 349, 1020, 381, 1340, 405, 1580, 561, 1932, 633, 2132, 881, 2512, 957, 2812, 1269, 3284, 1273, 3772, 1365, 4396, 1357, 4764, 1605, 5364, 1957, 5780, 2097, 6500, 2161, 6876, 2833, 7572, 2817

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

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=441; 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: A120073 A056894 A272145 * A334110 A117676 A085575

Adjacent sequences: A272218 A272219 A272220 * A272222 A272223 A272224

Keyword

nonn,easy

Author

Robert Price, Apr 22 2016

Status

approved