A277954 Decimal representation of the x-axis, from the left edge to the origin, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 14", based on the 5-celled von Neumann neighborhood.

1, 3, 6, 14, 26, 58, 106, 234, 426, 938, 1706, 3754, 6826, 15018, 27306, 60074, 109226, 240298, 436906, 961194, 1747626, 3844778, 6990506, 15379114, 27962026, 61516458, 111848106, 246065834, 447392426, 984263338, 1789569706, 3937053354, 7158278826

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 the 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

Index entries for sequences related to cellular automata

Index to 2D 5-Neighbor Cellular Automata

Index to Elementary Cellular Automata

Robert Price, Diagrams of the first 20 stages

Formula

Conjectures from Colin Barker, Nov 06 2016: (Start)

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

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

a(n) = (-8-(-2)^n+21*2^n)/12. (End)

Mathematica

CAStep[rule_, a_]:=Map[rule[[10-#]]&, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code=14; 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[1, i]], 2], {i, 1, stages-1}]
LinearRecurrence[{1, 4, -4}, {1, 3, 6}, 31] (* or *)
CoefficientList[ Series[(1 + 2x - x^2)/(1 - x - 4x^2 + 4x^3), {x, 0, 31}], x] (* Robert G. Wilson v, Nov 05 2016 *)

Crossrefs

Cf. A277952, A277953, A277955.

Sequence in context: A051749 A278788 A279828 * A279986 A279474 A365073

Adjacent sequences: A277951 A277952 A277953 * A277955 A277956 A277957

Keyword

nonn,easy

Author

Robert Price, Nov 05 2016

Status

approved