A284180 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 813", based on the 5-celled von Neumann neighborhood.

1, 1, 101, 101, 11101, 111101, 1110101, 11010101, 101111101, 111111101, 11111110101, 111111010101, 1111101111101, 11111111111101, 111111111110101, 1111111111010101, 11111111101111101, 111111111111111101, 1111111111111110101, 11111111111111010101

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 22 2017: (Start)

G.f.: (1 - 9*x + 91*x^2 - 909*x^3 + 10090*x^4 + 100*x^5 - 1000*x^6 - 90000*x^7 - 9000000*x^8 - 900000000*x^9 + 10000000000*x^10 + 1000000000*x^13 - 10000000000*x^14) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + x^2)).

a(n) = 10*a(n-1) + a(n-4) - 10*a(n-5) 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 = 813; 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. A284179, A284181, A284182.

Sequence in context: A279528 A283044 A284138 * A138719 A043639 A152054

Adjacent sequences: A284177 A284178 A284179 * A284181 A284182 A284183

Keyword

nonn,easy

Author

Robert Price, Mar 21 2017

Status

approved