A271885 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 406", based on the 5-celled von Neumann neighborhood.

1, 5, 8, 20, 20, 48, 37, 77, 45, 137, 84, 180, 120, 268, 136, 316, 133, 409, 225, 525, 268, 644, 276, 732, 344, 884, 420, 928, 485, 1117, 496, 1188, 556, 1460, 645, 1633, 724, 1792, 800, 1912, 844, 2164, 968, 2328, 1129, 2573, 1120, 2752, 1181, 3061, 1328

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=406; 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: A270168 A270984 A270321 * A295901 A143157 A275236

Adjacent sequences: A271882 A271883 A271884 * A271886 A271887 A271888

Keyword

nonn,easy

Author

Robert Price, Apr 16 2016

Status

approved