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%I #8 Jul 27 2024 03:54:07
%S 1,3,4,6,8,9,11,13,15,16,18,20,21,23,25,27,28,30,32,34,35,37,39,40,42,
%T 44,46,47,49,51,52,54,56,58,59,61,63,65,66,68,70,71,73,75,77,78,80,82,
%U 84,85,87,89,90,92,94,96,97,99,101,102,104,106,108,109
%N Positions of 0 in A285680.
%C A 3-way partition of the positive integers, by positions of 0, 1, 2 in A285680:
%C A285681: positions of 0; slope t = (15+sqrt(5))/10;
%C A285682: positions of 1; slope u = (4+sqrt(5);
%C A285683: positions of 2; slope v = (1+3*sqrt(5))/2;
%C where 1/t + 1/u + 1/v = 1. Conjecture: a(n) - a(n-1) is in {1,2} for n>=2.
%H Clark Kimberling, <a href="/A285681/b285681.txt">Table of n, a(n) for n = 1..10000</a>
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] ; (* A003849 *)
%t w = StringJoin[Map[ToString, s]]
%t w1 = StringReplace[w, {"1010" -> "2"}]
%t st = ToCharacterCode[w1] - 48; (* A285680 *)
%t Flatten[Position[st, 0]]; (* A285681 *)
%t Flatten[Position[st, 1]]; (* A285682 *)
%t Flatten[Position[st, 2]]; (* A285683 *)
%Y Cf. A003849, A285680, A285682, A285683.
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, May 11 2017
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