A283353 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 619", based on the 5-celled von Neumann neighborhood.

1, 2, 4, 14, 28, 62, 124, 254, 508, 1022, 2044, 4094, 8188, 16382, 32764, 65534, 131068, 262142, 524284, 1048574, 2097148, 4194302, 8388604, 16777214, 33554428, 67108862, 134217724, 268435454, 536870908, 1073741822, 2147483644, 4294967294, 8589934588

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 06 2017: (Start)

G.f.: (1 + 2*x)*(1 - 2*x + 3*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).

a(n) = 2^(n + 1) - 4 for n>0 and even.

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

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

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 619; 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. A283351, A283352.

Sequence in context: A115626 A116021 A288154 * A323656 A338740 A365544

Adjacent sequences: A283350 A283351 A283352 * A283354 A283355 A283356

Keyword

nonn,easy

Author

Robert Price, Mar 05 2017

Status

approved