A279962 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 221", based on the 5-celled von Neumann neighborhood.

1, 1, 2, 3, 28, 15, 112, 63, 448, 255, 1792, 1023, 7168, 4095, 28672, 16383, 114688, 65535, 458752, 262143, 1835008, 1048575, 7340032, 4194303, 29360128, 16777215, 117440512, 67108863, 469762048, 268435455, 1879048192, 1073741823, 7516192768, 4294967295

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, Dec 24 2016: (Start)

a(n) = 2^(n-3) * (5*(-1)^n + 9) for n>2 and even.

a(n) = (5*(-2)^n + 9*2^n - 8)/8 for n>2 and odd.

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

G.f.: (1 + x - 3*x^2 - 2*x^3 + 22*x^4 + 4*x^5 - 20*x^6) / ((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 = 221; 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. A279959, A279960, A279961.

Sequence in context: A279755 A279148 A279825 * A037320 A010344 A037316

Adjacent sequences: A279959 A279960 A279961 * A279963 A279964 A279965

Keyword

nonn,easy

Author

Robert Price, Dec 23 2016

Status

approved