A270091 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 78", based on the 5-celled von Neumann neighborhood.

1, 5, 8, 20, 20, 32, 48, 56, 88, 56, 96, 84, 140, 128, 180, 272, 260, 176, 312, 240, 328, 280, 408, 400, 552, 308, 536, 476, 748, 560, 848, 784, 1064, 768, 1092, 828, 1044, 928, 1068, 1196, 1100, 1124, 1332, 1408, 1660, 1472, 1560, 1716, 2180, 1684, 2056

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=78; 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: A271095 A194266 A169703 * A270733 A270168 A270984

Adjacent sequences: A270088 A270089 A270090 * A270092 A270093 A270094

Keyword

nonn,easy

Author

Robert Price, Mar 10 2016

Status

approved