A288827 Decimal representation of the diagonal from the corner to the origin 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, 13, 27, 59, 119, 247, 495, 1007, 2015, 4063, 8127, 16319, 32639, 65407, 130815, 261887, 523775, 1048063, 2096127, 4193279, 8386559, 16775167, 33550335, 67104767, 134209535, 268427263, 536854527, 1073725439, 2147450879, 4294934527, 8589869055

Offset

0,2

Comments

Initialized with a single black (ON) cell at stage zero.

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 Media, 2002; p. 170.

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

Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Formula

From Stefano Spezia, May 25 2022: (Start)

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

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

a(n) = A000225(n+1) - A016116(n) for n > 1. (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, A288828.

Cf. A000225, A016116.

Sequence in context: A141630 A289467 A289533 * A084173 A354951 A223645

Adjacent sequences: A288824 A288825 A288826 * A288828 A288829 A288830

Keyword

nonn,easy

Author

Robert Price, Jun 17 2017

Status

approved