A290195 Decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 705", based on the 5-celled von Neumann neighborhood.

1, 1, 5, 11, 7, 47, 31, 191, 127, 767, 511, 3071, 2047, 12287, 8191, 49151, 32767, 196607, 131071, 786431, 524287, 3145727, 2097151, 12582911, 8388607, 50331647, 33554431, 201326591, 134217727, 805306367, 536870911, 3221225471, 2147483647, 12884901887

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

Conjecture: For odd n > 3, a(n) = 2^(n-1) - 1, for even n > 3, a(n) = 3*2^(n-1) - 1. - David A. Corneth, Jul 23 2017

From Chai Wah Wu, Nov 01 2018: (Start)

a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) for n > 5 (conjectured).

G.f.: (16*x^5 - 20*x^4 + 6*x^3 + 1)/((x - 1)*(2*x - 1)*(2*x + 1)) (conjectured). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 705; 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]], 10], {i, 1, stages - 1}]

Crossrefs

Cf. A290192, A290193, A290194.

Sequence in context: A368055 A358613 A329005 * A082952 A113964 A356480

Adjacent sequences: A290192 A290193 A290194 * A290196 A290197 A290198

Keyword

nonn,easy

Author

Robert Price, Jul 23 2017

Status

approved