A272217 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 438", based on the 5-celled von Neumann neighborhood.

1, 5, 8, 20, 32, 40, 36, 100, 113, 112, 140, 192, 224, 244, 256, 352, 380, 440, 472, 528, 540, 552, 588, 700, 716, 876, 912, 1116, 1048, 1052, 1076, 1448, 1433, 1560, 1500, 1853, 1836, 1924, 1808, 2132, 2184, 2396, 2224, 2504, 2756, 2692, 2733, 3232, 2881

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=438; 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: A270022 A271195 A270630 * A084568 A340509 A198635

Adjacent sequences: A272214 A272215 A272216 * A272218 A272219 A272220

Keyword

nonn,easy

Author

Robert Price, Apr 22 2016

Status

approved