A283849 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 734", based on the 5-celled von Neumann neighborhood.

1, 11, 111, 1111, 11101, 111111, 1110111, 11111111, 111011101, 1111111111, 11101111011, 111111111011, 1110111011111, 11111111111111, 111011110110111, 1111111111111111, 11101110111111111, 111111111111111111, 1110111101101111111, 11111111111111111111

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 Chai Wah Wu, May 06 2024: (Start)

a(n) = a(n-1) + 10000*a(n-4) - 10000*a(n-5) for n > 25.

G.f.: (-1000000000000*x^25 + 1000000000000*x^24 - 100000000000*x^23 + 100000000000*x^22 + 100000000*x^21 - 100000000*x^20 - 1000000*x^16 + 1001000*x^15 - 1000*x^14 + 100*x^12 - 100*x^10 + 10*x^9 - 10*x^8 + 1000*x^7 - 1000*x^6 + 10*x^5 - 10*x^4 + 1000*x^3 + 100*x^2 + 10*x + 1)/(10000*x^5 - 10000*x^4 - x + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 734; 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. A283850, A283864, A283865.

Sequence in context: A284020 A284087 A284419 * A283604 A090103 A061851

Adjacent sequences: A283846 A283847 A283848 * A283850 A283851 A283852

Keyword

nonn,easy

Author

Robert Price, Mar 17 2017

Status

approved