A273557 Number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 785", based on the 5-celled von Neumann neighborhood.

1, 4, 13, 28, 49, 77, 104, 141, 180, 241, 268, 353, 392, 485, 544, 629, 700, 825, 884, 1033, 1104, 1261, 1352, 1501, 1604, 1777, 1868, 2049, 2184, 2341, 2464, 2645, 2780, 3033, 3156, 3433, 3568, 3853, 4008, 4285, 4452, 4753, 4908, 5217, 5416, 5701, 5888

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=785; 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: A056107 A155433 A272746 * A272790 A273564 A155392

Adjacent sequences: A273554 A273555 A273556 * A273558 A273559 A273560

Keyword

nonn,easy

Author

Robert Price, May 25 2016

Status

approved