A281282 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 350", based on the 5-celled von Neumann neighborhood.

1, 11, 11, 1011, 11011, 1011, 111011, 11001011, 1111011, 100001011, 10110111011, 111111001011, 1111011, 1000100001011, 101010110111011, 1111010011001011, 11011101111011, 11001000100001011, 1001101010110111011, 11100101010011001011, 110111011101111011

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 05 2024: (Start)

a(n) = 10001*a(n-4) - 10000*a(n-8) for n > 23.

G.f.: (9900000000000000*x^23 + 9010000000000000*x^22 - 1000000000000000*x^21 - 11000000000000*x^20 - 9999000000000000*x^19 - 9000000000000000*x^18 + 1000000000000000*x^17 + 11011100000000*x^16 - 101100000000*x^15 - 100000000000*x^14 - 11000000000*x^12 + 1100000000*x^11 + 9000000000*x^10 + 90000000*x^9 - 109000000*x^8 + 890000*x^7 + 1000*x^6 - 109000*x^5 + 1010*x^4 + 1011*x^3 + 11*x^2 + 11*x + 1)/(10000*x^8 - 10001*x^4 + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 350; 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. A281281, A281283, A281284.

Sequence in context: A283218 A283136 A280562 * A281637 A281099 A092805

Adjacent sequences: A281279 A281280 A281281 * A281283 A281284 A281285

Keyword

nonn,easy

Author

Robert Price, Jan 18 2017

Status

approved