A273560 First differences of 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.

3, 9, 15, 21, 28, 27, 37, 39, 61, 27, 85, 39, 93, 59, 85, 71, 125, 59, 149, 71, 157, 91, 149, 103, 173, 91, 181, 135, 157, 123, 181, 135, 253, 123, 277, 135, 285, 155, 277, 167, 301, 155, 309, 199, 285, 187, 309, 199, 365, 187, 373, 231, 341, 235, 341, 247

Offset

0,1

Comments

Initialized with a single black (ON) cell at stage zero.

First negative term is a(124) = -83. - Georg Fischer, Feb 15 2019

References

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

Robert Price, Table of n, a(n) for n = 0..127

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}];
on=Map[Function[Apply[Plus, Flatten[#1]]], ca] (* Count ON cells at each stage *)
Table[on[[i+1]]-on[[i]], {i, 1, Length[on]-1}] (* Difference at each stage *)

Crossrefs

Cf. A273557.

Sequence in context: A162843 A102954 A272749 * A246388 A272793 A273567

Adjacent sequences: A273557 A273558 A273559 * A273561 A273562 A273563

Keyword

sign,easy

Author

Robert Price, May 25 2016

Status

approved