A288709 Positions of 1's in A288707; complement of A288708.

3, 7, 13, 17, 23, 29, 33, 39, 43, 49, 55, 59, 65, 71, 75, 81, 85, 91, 97, 101, 107, 111, 117, 123, 127, 133, 139, 143, 149, 153, 159, 165, 169, 175, 181, 185, 191, 195, 201, 207, 211, 217, 221, 227, 233, 237, 243, 249, 253, 259, 263, 269, 275, 279, 285, 289

Offset

1,1

Comments

Conjecture: a(n)/n-> 3 + sqrt(5), and if m denotes this number, then -1 < m - a(n)/n) < 3 for n >= 1.

From Michel Dekking, Oct 19 2018: (Start)

Here is a proof of this conjecture. Note that

q(n) := n/a(n) = [a(1)+a(2)+...+a(n)]/a(n)

is the frequency of 1's in the first a(n) terms of the sequence.

It follows from the sequel that (q(n)) converges as n->infinity.

So we have to show that

q(n) --> 1/m = (3-sqrt(5))/4.

It is useful here to profit from Kimberling's observation in the Comments of x:=A288707 that x is the {0->00, 1->10} transform of the morphism 0->10, 1->0.

We see from this that a 1 occurs at position 2k-1 in x if

and only if a 1 occurs at position k in the fixed point

y = A189661 = 0, 1, 0, 1, 0, 0, 1, 0, 1, 0,...

with y(1)=0 of the square of the time reversed Fibonacci morphism 0->10,1->0 (this explains why all the numbers in (a(n)) are odd).

Using the incidence matrix of the morphism 0->010, 1->10,

one can calculate that the frequency of 1's in y equals (3-sqrt(5))/2,

and so the frequency of 1's in x is half this number, and we have proved that

q(n) --> (3-sqrt(5))/4.

The bounds -1 < m - a(n)/n < 3 are equivalent to bounds on

2q(n)-(3-sqrt(5))/2.

The latter can be proved by checking them for a finite number of n, and then using the exponential convergence of the 2q(n) to (3-sqrt(5))/2 (a consequence of the Perron-Frobenius theorem). (End)

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

Mathematica

s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "00"}]
Table[w[n], {n, 0, 8}]
st = ToCharacterCode[w[10]] - 48   (* A288707 *)
Flatten[Position[st, 0]]  (* A288708 *)
Flatten[Position[st, 1]]  (* A288709 *)

Crossrefs

Cf. A288707, A288708.

Sequence in context: A285669 A063239 A063226 * A040106 A378429 A191028

Adjacent sequences: A288706 A288707 A288708 * A288710 A288711 A288712

Keyword

nonn,easy

Author

Clark Kimberling, Jun 16 2017

Status

approved