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A285679 Positions of 2 in A285677. 4

%I

%S 3,5,10,12,17,22,24,29,31,36,41,43,48,53,55,60,62,67,72,74,79,81,86,

%T 91,93,98,103,105,110,112,117,122,124,129,134,136,141,143,148,153,155,

%U 160,162,167,172,174,179,184,186,191,193,198,203,205,210,212,217

%N Positions of 2 in A285677.

%C A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285677:

%C A285678: positions of 0; slope t = (4+sqrt(5))/2;

%C A182761: positions of 1; slope u = (7-sqrt(5))/2;

%C A285679: positions of 2; slope v = (1+3*sqrt(5))/2;

%C where 1/t + 1/u + 1/v = 1.

%C Conjecture: a(n) - a(n-1) is in {2,5} for n>=2.

%C See A285683 for a proof of this conjecture. - _Michel Dekking_, Oct 09 2018

%C a(n) = A285683(n-1) for n>1, see A285683 for a proof. - _Michel Dekking_, Oct 09 2018

%H Clark Kimberling, <a href="/A285679/b285679.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = 3*floor((n-1)*phi) - n + 4

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)

%t w = StringJoin[Map[ToString, s]]

%t w1 = StringReplace[w, {"0010" -> "2"}]

%t st = ToCharacterCode[w1] - 48; (* A285677 *)

%t Flatten[Position[st, 0]]; (* A285678 *)

%t Flatten[Position[st, 1]]; (* A182761 *)

%t Flatten[Position[st, 2]]; (* A285679 *)

%Y Cf. A003849, A284620, A285678, A182761, A285679.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, May 11 2017

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Last modified March 19 17:21 EDT 2019. Contains 321330 sequences. (Running on oeis4.)