A272938 First differences of number of active (ON, black) cells in n-th stage of growth of two-dimensional cellular automaton defined by "Rule 561", based on the 5-celled von Neumann neighborhood.

3, 13, 11, 33, 28, 32, 39, 73, 44, 80, 60, 72, 44, 92, 39, 181, 80, 136, 192, 72, 128, 104, 80, 240, 180, 136, 88, 172, 128, 180, 63, 389, 192, 304, 212, 220, 216, 96, 196, 244, 376, 172, 352, 268, 180, 264, 16, 404, 460, 344, 272, 452, 164, 256, 156, 568

Offset

0,1

Comments

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

The first negative term is a(77) = -48. - 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=561; 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. A272769.

Sequence in context: A272849 A273613 A212552 * A273649 A272994 A273674

Adjacent sequences: A272935 A272936 A272937 * A272939 A272940 A272941

Keyword

sign,easy

Author

Robert Price, May 11 2016

Status

approved