A284353 Decimal 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 899", based on the 5-celled von Neumann neighborhood.

1, 1, 7, 13, 31, 61, 127, 253, 511, 1021, 2047, 4093, 8191, 16381, 32767, 65533, 131071, 262141, 524287, 1048573, 2097151, 4194301, 8388607, 16777213, 33554431, 67108861, 134217727, 268435453, 536870911, 1073741821, 2147483647, 4294967293, 8589934591

Offset

0,3

Comments

Initialized with a single black (ON) cell at stage zero.

If one begins the Generalized Jacobsthal numbers (A083579) with a(0)=1, instead of a(0)=0, the same sequence will be obtained. - Henrik Lipskoch, Jan 28 2021

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 26 2017: (Start)

G.f.: (1 - x + 4*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).

a(n) = 2^(n+1) - 1 for n even.

a(n) = 2^(n+1) - 3 for n odd.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2. (End)

Conjecture: For n > 0, a(n) = Sum_{k=0..n-1} C(n,k) * (2-(-1)^k). - Wesley Ivan Hurt, Sep 23 2017

Apparently, a(n) = 6*A000975(n-1) + 1 for n >= 1. - Hugo Pfoertner, Jan 28 2021

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 899; 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]], 2], {i, 1, stages - 1}]

Crossrefs

Cf. A000975, A284351, A284352, A284354.

Sequence in context: A272204 A107146 A201601 * A129781 A174217 A040061

Adjacent sequences: A284350 A284351 A284352 * A284354 A284355 A284356

Keyword

nonn,easy

Author

Robert Price, Mar 25 2017

Status

approved