A288828 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 515", based on the 5-celled von Neumann neighborhood.

1, 3, 5, 11, 27, 55, 119, 239, 495, 991, 2015, 4031, 8127, 16255, 32639, 65279, 130815, 261631, 523775, 1047551, 2096127, 4192255, 8386559, 16773119, 33550335, 67100671, 134209535, 268419071, 536854527, 1073709055, 2147450879, 4294901759, 8589869055

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

From Chai Wah Wu, Sep 06 2019: (Start)

a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4) for n > 5 (conjectured).

G.f.: (8*x^5 - 8*x^4 - 2*x^3 + 4*x^2 - 1)/((x - 1)*(2*x - 1)*(2*x^2 - 1)) (conjectured). (End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 515; 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. A288825, A288826, A288827.

Sequence in context: A289468 A289534 A032364 * A088357 A369344 A204857

Adjacent sequences: A288825 A288826 A288827 * A288829 A288830 A288831

Keyword

nonn,easy

Author

Robert Price, Jun 17 2017

Status

approved