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A285679 Positions of 2 in A285677.

%I A285679 #12 Oct 14 2018 05:48:43
%S A285679 3,5,10,12,17,22,24,29,31,36,41,43,48,53,55,60,62,67,72,74,79,81,86,
%T A285679 91,93,98,103,105,110,112,117,122,124,129,134,136,141,143,148,153,155,
%U A285679 160,162,167,172,174,179,184,186,191,193,198,203,205,210,212,217
%N A285679 Positions of 2 in A285677.
%C A285679 A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:
%C A285679 A285678: positions of 0; slope t = (4+sqrt(5))/2;
%C A285679 A182761: positions of 1; slope u = (7-sqrt(5))/2;
%C A285679 A285679: positions of 2; slope v = (1+3*sqrt(5))/2;
%C A285679 where 1/t + 1/u + 1/v = 1.
%C A285679 Conjecture:  a(n) - a(n-1) is in {2,5} for n>=2.
%C A285679 See A285683 for a proof of this conjecture. - _Michel Dekking_, Oct 09 2018
%C A285679 a(n) = A285683(n-1)  for n>1, see A285683 for a proof. - _Michel Dekking_, Oct 09 2018
%H A285679 Clark Kimberling, <a href="/A285679/b285679.txt">Table of n, a(n) for n = 1..10000</a>
%F A285679 a(n) = 3*floor((n-1)*phi) - n + 4
%t A285679 s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
%t A285679 w = StringJoin[Map[ToString, s]]
%t A285679 w1 = StringReplace[w, {"0010" -> "2"}]
%t A285679 st = ToCharacterCode[w1] - 48; (* A285677 *)
%t A285679 Flatten[Position[st, 0]];  (* A285678 *)
%t A285679 Flatten[Position[st, 1]];  (* A182761 *)
%t A285679 Flatten[Position[st, 2]];  (* A285679 *)
%Y A285679 Cf. A003849, A284620, A285678, A182761, A285679.
%K A285679 nonn,easy
%O A285679 1,1
%A A285679 _Clark Kimberling_, May 11 2017