A285612 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 62", based on the 5-celled von Neumann neighborhood.

1, 1, 10, 10, 110, 110, 1110, 1110, 11110, 11110, 111110, 111110, 1111110, 1111110, 11111110, 11111110, 111111110, 111111110, 1111111110, 1111111110, 11111111110, 11111111110, 111111111110, 111111111110, 1111111111110, 1111111111110, 11111111111110

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, Apr 23 2017: (Start)

G.f.: (1 - x^2 + 10*x^4) / ((1 - x)*(1 - 10*x^2)).

a(n) = 10*(10^(n/2) - 1)/9 for n>1 and even.

a(n) = (10^((n+1)/2) - 10)/9 for n>1 and odd.

a(n) = a(n-1) + 10*a(n-2) - 10*a(n-3) for n>2.

(End)

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 62; 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. A285613, A056453, A233411.

Sequence in context: A288300 A286407 A285608 * A287600 A047817 A065243

Adjacent sequences: A285609 A285610 A285611 * A285613 A285614 A285615

Keyword

nonn,easy

Author

Robert Price, Apr 22 2017

Status

approved