A284545 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 998", based on the 5-celled von Neumann neighborhood.

1, 11, 101, 1011, 10101, 101011, 1010101, 10101011, 101010101, 1010101011, 10101010101, 101110101011, 1011101010101, 10110010101011, 101111101010101, 1011110110101011, 10111110101010101, 101111111010101011, 1011111110101010101, 10111111101110101011

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) = 10*a(n-1) + a(n-8) - 10*a(n-9) for n > 25.

G.f.: (90000000000*x^25 + 81000000000*x^24 - 8000000000*x^23 + 9988900000000*x^22 - 897900000000*x^21 + 89900000000*x^20 + 10000000000*x^17 + 9000000000*x^16 - 900000000*x^15 + 11000000000*x^14 - 1000000000*x^13 + 100000000*x^11 - 10*x^8 + x^7 - 9*x^6 + x^5 - 9*x^4 + x^3 - 9*x^2 + x + 1)/(10*x^9 - x^8 - 10*x + 1). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 998; 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. A284544, A284546, A284547.

Sequence in context: A284537 A266508 A284424 * A283645 A283713 A283601

Adjacent sequences: A284542 A284543 A284544 * A284546 A284547 A284548

Keyword

nonn,easy

Author

Robert Price, Mar 28 2017

Status

approved