A273687 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 846", based on the 5-celled von Neumann neighborhood.

1, 5, 9, 17, 17, 37, 53, 49, 73, 97, 89, 109, 129, 181, 241, 261, 297, 245, 309, 325, 401, 425, 477, 557, 565, 581, 613, 681, 789, 857, 761, 1081, 981, 1049, 1193, 1097, 1229, 1213, 1305, 1545, 1513, 1569, 1677, 1737, 1837, 1857, 1913, 2121, 2065, 2233, 2353

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=846; 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: A272742 A273534 A273115 * A102179 A345920 A315112

Adjacent sequences: A273684 A273685 A273686 * A273688 A273689 A273690

Keyword

nonn,easy

Author

Robert Price, May 27 2016

Status

approved