A282655 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 491", based on the 5-celled von Neumann neighborhood.

1, 2, 7, 2, 31, 2, 127, 34, 511, 130, 2047, 674, 8191, 2690, 32767, 10914, 131071, 43650, 524287, 174754, 2097151, 699010, 8388607, 2796194, 33554431, 11184770, 134217727, 44739234, 536870911, 178956930, 2147483647, 715827874, 8589934591, 2863311490

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, Feb 20 2017: (Start)

a(n) = 4*a(n-2) + a(n-4) - 4*a(n-6) for n>5.

G.f.: (1 + 2*x + 3*x^2 - 6*x^3 + 2*x^4 - 8*x^5 + 32*x^7 + 128*x^11) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)*(1 + x^2)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 491; 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. A282652, A282653, A282654.

Sequence in context: A100489 A176379 A282454 * A282803 A341093 A248683

Adjacent sequences: A282652 A282653 A282654 * A282656 A282657 A282658

Keyword

nonn,easy

Author

Robert Price, Feb 20 2017

Status

approved