A286770 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 221", based on the 5-celled von Neumann neighborhood.

1, 10, 0, 1110, 1, 111110, 1, 11111110, 1, 1111111110, 1, 111111111110, 1, 11111111111110, 1, 1111111111111110, 1, 111111111111111110, 1, 11111111111111111110, 1, 1111111111111111111110, 1, 111111111111111111111110, 1, 11111111111111111111111110, 1

Offset: 0

Author

Robert Price, May 14 2017

Keywords

• nonn
• easy

Comments

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

a(n) = A280410(n) for n >= 3 ? - Michel Marcus, May 17 2017

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

Crossrefs

Cf. A286771, A286772, A286773, A280410.

Programs

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0},{2, 1, 2}, {0, 2, 0}}, a, 2],{2}];
code = 221; 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}]

Formula

Conjectures from Colin Barker, May 14 2017: (Start)

G.f.: (1 + 10*x - 101*x^2 + 100*x^3 + 101*x^4 - 100*x^6) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).

a(n) = 1 for n>2 and even.

a(n) = 10*(10^n - 1)/9 for n>2 and odd.

a(n) = 101*a(n-2) - 100*a(n-4) for n>4.

(End)