A279030 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 129", based on the 5-celled von Neumann neighborhood.

1, 0, 5, 0, 21, 0, 85, 0, 341, 0, 1365, 0, 5461, 0, 21845, 0, 87381, 0, 349525, 0, 1398101, 0, 5592405, 0, 22369621, 0, 89478485, 0, 357913941, 0, 1431655765, 0, 5726623061, 0, 22906492245, 0, 91625968981, 0, 366503875925, 0, 1466015503701, 0, 5864062014805

Offset

0,3

Comments

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

Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 129", based on the 5-celled von Neumann neighborhood. Initialized with a single black (ON) cell at stage zero.

A279030 is related to A000975 in such a way that a(2*n) = b(2*n + 1) and a(2*n + 1) = b(2*n + 2) - 2*b(2*n + 1) = b(2*n + 2) - 2*a(2*n), where b(2*n + 1) and b(2*n + 2) are members of A000975. - Mario C. Enriquez, Dec 07 2016

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, Dec 04 2016: (Start)

a(n) = (1 + (-1)^n)*(2^(2 + n)-1)/6.

a(n) = 5*a(n-2) - 4*a(n-4) for n>3.

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

(End)

Bisection appears to be A002450 ((4^n-1)/3). - N. J. A. Sloane, Dec 06 2016

Mathematica

CAStep[rule_, a_] := Map[rule[[10 - #]] &, ListConvolve[{{0, 2, 0}, {2, 1, 2}, {0, 2, 0}}, a, 2], {2}];
code = 129; 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}]

Crossrefs

Cf. A000975, A002450, A279028.

Sequence in context: A056461 A167355 A282577 * A176868 A279143 A279144

Adjacent sequences: A279027 A279028 A279029 * A279031 A279032 A279033

Keyword

nonn,easy

Author

Robert Price, Dec 03 2016

Status

approved