A288709 - OEIS

%I #11 Oct 19 2018 09:54:10

%S 3,7,13,17,23,29,33,39,43,49,55,59,65,71,75,81,85,91,97,101,107,111,

%T 117,123,127,133,139,143,149,153,159,165,169,175,181,185,191,195,201,

%U 207,211,217,221,227,233,237,243,249,253,259,263,269,275,279,285,289

%N Positions of 1's in A288707; complement of A288708.

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

%C From _Michel Dekking_, Oct 19 2018: (Start)

%C Here is a proof of this conjecture.  Note that

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

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

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

%C So we have to show  that

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

%C 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.

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

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

%C    y = A189661 = 0, 1, 0, 1, 0, 0, 1, 0, 1, 0,...

%C 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).

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

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

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

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

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

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

%C 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)

%H Clark Kimberling, <a href="/A288709/b288709.txt">Table of n, a(n) for n = 1..10000</a>

%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "00"}]

%t Table[w[n], {n, 0, 8}]

%t st = ToCharacterCode[w[10]] - 48   (* A288707 *)

%t Flatten[Position[st, 0]]  (* A288708 *)

%t Flatten[Position[st, 1]]  (* A288709 *)

%Y Cf. A288707, A288708.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jun 16 2017