A283585 Binary representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 654", based on the 5-celled von Neumann neighborhood.

1, 11, 111, 1101, 11101, 110101, 1110001, 11011011, 111011011, 1101011011, 11100011011, 110110111011, 1110110101011, 11010110101011, 111000110101011, 1101101110101011, 11101101110101011, 110101101110101011, 1110001101110101011, 11011011101110101011

Offset

0,2

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 12 2017: (Start)

G.f.: (1 + 10*x + 100*x^2 + 990*x^3 + 10000*x^4 + 99000*x^5 + 999900*x^6 + 9901010*x^7 - 10000000*x^9 - 1000000*x^10 + 10100000*x^11 - 10000*x^12) / ((1 - x)*(1 - 10*x)*(1 + 10*x)*(1 + 100*x^2)*(1 + 10000*x^4)).

a(n) = a(n-1) + 100000000*a(n-8) - 100000000*a(n-9) for n>12.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 654; 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[1, i]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A283586, A283587, A283588.

Sequence in context: A283175 A284274 A185000 * A283703 A284399 A284540

Adjacent sequences: A283582 A283583 A283584 * A283586 A283587 A283588

Keyword

nonn,easy

Author

Robert Price, Mar 11 2017

Status

approved