A282802 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 507", based on the 5-celled von Neumann neighborhood.

1, 1, 7, 4, 31, 20, 127, 84, 511, 340, 2047, 1364, 8191, 5460, 32767, 21844, 131071, 87380, 524287, 349524, 2097151, 1398100, 8388607, 5592404, 33554431, 22369620, 134217727, 89478484, 536870911, 357913940, 2147483647, 1431655764, 8589934591, 5726623060

Offset

0,3

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 22 2017: (Start)

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

a(n) = 2*(2^n - 2) / 3 for n>1 and odd.

a(n) = 5*a(n-2) - 4*a(n-4) for n>3.

G.f.: (1 + x + 2*x^2 - x^3 + 4*x^5) / ((1 - x)*(1 + x)*(1 - 2*x)*(1 + 2*x)).

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 507; 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. A282800, A282801, A282803.

Sequence in context: A282453 A282654 A282798 * A248278 A270235 A270719

Adjacent sequences: A282799 A282800 A282801 * A282803 A282804 A282805

Keyword

nonn,easy

Author

Robert Price, Feb 21 2017

Status

approved