A288701 Binary representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 497", based on the 5-celled von Neumann neighborhood.

1, 10, 11, 1110, 11, 111100, 1111, 11111000, 1111, 1111111000, 1111, 111111111000, 1111, 11111111111000, 11111111, 1111111110011110, 11110011, 111111111110011110, 11110011, 11111111111110011100, 1111111, 1111111111111110011000, 11111111

Offset

0,2

Comments

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

Is this (apart from a(1)) the same as A276540? - R. J. Mathar, Jul 19 2024

References

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

Links

Robert Price, Table of n, a(n) for n = 0..126

Robert Price, Diagrams of 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

Wolfram Research, Wolfram Atlas of Simple Programs

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 = 497; 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}];
Table[FromDigits[Part[ca[[i]] [[i]], Range[i, 2 * i - 1]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A288702, A288704, A288705.

Sequence in context: A281752 A282117 A287094 * A282367 A282002 A282656

Adjacent sequences: A288698 A288699 A288700 * A288702 A288703 A288704

Keyword

nonn,easy

Author

Robert Price, Jun 13 2017

Status

approved