A287129 Binary 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 245", based on the 5-celled von Neumann neighborhood.

1, 10, 0, 1110, 11, 111110, 11, 11111100, 11, 1111111100, 11, 111111111100, 11, 11111111111100, 11, 1111111111111100, 11, 111111111111111100, 11, 11111111111111111100, 11, 1111111111111111111100, 11, 111111111111111111111100, 11, 11111111111111111111111100

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

Conjectures from Colin Barker, May 20 2017: (Start)

G.f.: (1 + 10*x - 101*x^2 + 100*x^3 + 111*x^4 - 1100*x^6 - 10*x^7 + 1000*x^9) / ((1 - x)*(1 + x)*(1 - 10*x)*(1 + 10*x)).

a(n) = 11 for n>5 and even.

a(n) = 10*(10^n - 10)/9 for n>5 and odd.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 245; 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. A287130, A287131, A287132.

Sequence in context: A286770 A286503 A286403 * A287190 A277926 A222521

Adjacent sequences: A287126 A287127 A287128 * A287130 A287131 A287132

Keyword

nonn,easy

Author

Robert Price, May 20 2017

Status

approved