A284354 Decimal 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 899", based on the 5-celled von Neumann neighborhood.

1, 2, 7, 11, 31, 47, 127, 191, 511, 767, 2047, 3071, 8191, 12287, 32767, 49151, 131071, 196607, 524287, 786431, 2097151, 3145727, 8388607, 12582911, 33554431, 50331647, 134217727, 201326591, 536870911, 805306367, 2147483647, 3221225471, 8589934591

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 26 2017: (Start)

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

a(n) = (2^(n+3) - 4)/4 for n even.

a(n) = (3*2^(n+1) - 4)/4 for n odd.

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n>2.

(End)

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[i, 2 * i - 1]], 2], {i , 1, stages - 1}]

Crossrefs

Cf. A284351, A284352, A284353.

Sequence in context: A361151 A327552 A336342 * A228076 A123151 A371201

Adjacent sequences: A284351 A284352 A284353 * A284355 A284356 A284357

Keyword

nonn,easy

Author

Robert Price, Mar 25 2017

Status

approved