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A288709 Positions of 1's in A288707; complement of A288708. 5
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 (list; graph; refs; listen; history; text; internal format)
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 A191028 A034914

Adjacent sequences:  A288706 A288707 A288708 * A288710 A288711 A288712

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 16 2017

STATUS

approved

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Last modified February 22 16:36 EST 2020. Contains 332140 sequences. (Running on oeis4.)