%I #27 Oct 11 2017 10:34:01
%S 1,2,3,5,6,7,8,10,11,12,14,15,16,17,19,20,21,22,24,25,26,28,29,30,31,
%T 33,34,35,37,38,39,40,42,43,44,45,47,48,49,51,52,53,54,56,57,58,59,61,
%U 62,63,65,66,67,68,70,71,72,74,75,76,77,79,80,81,82,84
%N Positions of 1 in A287769; complement of A276855.
%C Conjecture: 0 < n*r - a(n) < 1 for n >= 1, where r = (15 - sqrt(5))/10. It has been verified by computer that a(n) = floor(n*r) for n=1..3*10^6.
%C This conjecture can be proved from the result in the Comments of A287769, where it is shown that A287769 is a Sturmian sequence with slope s := 1 - 1/(3+phi) = (15+sqrt(5))/22. The conjecture then follows from Lemma 9.1.3 in "Automatic sequences", since r = 1/s. - _Michel Dekking_, Oct 11 2017
%D Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Theory, applications, generalizations, Cambridge University Press, Cambridge, 2003, xvi+571.
%H Clark Kimberling, <a href="/A287770/b287770.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = floor(n*r), where r = (15 - sqrt(5))/10. - _Michel Dekking_, Oct 11 2017
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)
%t w = StringJoin[Map[ToString, s]]
%t w1 = StringReplace[w, {"0" -> "1", "1" -> "110"}]
%t st = ToCharacterCode[w1] - 48 (* A287769 *)
%t Flatten[Position[st, 0]] (* A276855 *)
%t Flatten[Position[st, 1]] (* A287770 *)
%Y Cf. A276855, A287769.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jun 03 2017