A270160 Number of active (ON,black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 105", based on the 5-celled von Neumann neighborhood.

1, 4, 9, 32, 13, 92, 45, 156, 36, 309, 56, 404, 85, 624, 177, 764, 125, 1084, 181, 1268, 209, 1576, 321, 1800, 277, 2204, 313, 2584, 336, 2957, 532, 3245, 456, 3941, 504, 4289, 576, 4880, 749, 5284, 657, 6012, 820, 6516, 833, 7076, 1185, 7560, 1025, 8476

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=105; 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: A270909 A270152 A271305 * A271414 A270177 A268277

Adjacent sequences: A270157 A270158 A270159 * A270161 A270162 A270163

Keyword

nonn,easy

Author

Robert Price, Mar 12 2016

Status

approved