A283348 Binary representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 617", based on the 5-celled von Neumann neighborhood.

1, 0, 110, 1000, 1110, 11000, 111110, 1111000, 11111110, 111111000, 1111111110, 11111111000, 111111111110, 1111111111000, 11111111111110, 111111111111000, 1111111111111110, 11111111111111000, 111111111111111110, 1111111111111111000, 11111111111111111110

Offset

0,3

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..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

Formula

Conjectures from Colin Barker, Mar 06 2017: (Start)

G.f.: (1 - 10*x + 109*x^2 - 90*x^3 - 9000*x^4 + 10000*x^6) / ((1 - x)*(1 + x)*(1 - 10*x)).

a(n) = (10^n - 10) / 9 for n>3 and even.

a(n) = (10^n - 1000) / 9 for n>3 and odd.

a(n) = 10*a(n-1) + a(n-2) - 10*a(n-3) for n>4.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 617; 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. A282871, A283349, A283350.

Sequence in context: A283284 A283373 A283401 * A284236 A284300 A284348

Adjacent sequences: A283345 A283346 A283347 * A283349 A283350 A283351

Keyword

nonn,easy

Author

Robert Price, Mar 05 2017

Status

approved