A285682 - OEIS

%I #8 Jul 27 2024 03:55:34

%S 2,7,14,19,26,33,38,45,50,57,64,69,76,83,88,95,100,107,114,119,126,

%T 131,138,145,150,157,164,169,176,181,188,195,200,207,214,219,226,231,

%U 238,245,250,257,262,269,276,281,288,295,300,307,312,319,326,331,338

%N Positions of 1 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 {5,7} for n>=2.

%H Clark Kimberling, <a href="/A285682/b285682.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, A285681, A285683.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, May 11 2017