A273250 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 619", based on the 5-celled von Neumann neighborhood.

1, 5, 17, 37, 65, 101, 137, 181, 241, 309, 377, 453, 537, 629, 705, 789, 913, 1045, 1177, 1317, 1465, 1621, 1761, 1909, 2089, 2277, 2449, 2629, 2817, 3013, 3145, 3285, 3537, 3797, 4057, 4325, 4601, 4885, 5153, 5429, 5737, 6053, 6353, 6661, 6977, 7301, 7561

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

Robert Price, Diagrams of the first 20 stages

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=619; 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: A273212 A273274 A357058 * A053755 A162373 A146781

Adjacent sequences: A273247 A273248 A273249 * A273251 A273252 A273253

Keyword

nonn,easy

Author

Robert Price, May 18 2016

Status

approved