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A003157 A self-generating sequence (see Comments in A003156 for the definition).
(Formerly M2713)
10

%I M2713

%S 3,8,11,14,19,24,29,32,35,40,43,46,51,54,57,62,67,72,75,78,83,88,93,

%T 96,99,104,109,114,117,120,125,128,131,136,139,142,147,152,157,160

%N A self-generating sequence (see Comments in A003156 for the definition).

%C Indices of c in the sequence closed under a -> abc, b -> a, c -> a, starting with a(1) = a; see A092606 where a = 0, b = 2, c = 1. - _Philippe Deléham_, Apr 12 2004

%C These are the positions of 1 in A286044; complement of A286045; conjecture: a(n)/n -> 4. - _Clark Kimberling_, May 07 2017

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Clark Kimberling, <a href="/A003157/b003157.txt">Table of n, a(n) for n = 1..10000</a>

%H L. Carlitz, R. Scoville, and V. E. Hoggatt, Jr., <a href="http://www.fq.math.ca/Scanned/10-5/carlitz3-a.pdf">Representations for a special sequence</a>, Fibonacci Quarterly 10.5 (1972), 499-518, 550.

%F Numbers n such that A003159(n) is even. a(n) = A003158(n) + 1 = A036554(n) + n. - _Philippe Deléham_, Feb 22 2004

%e As a word, A286044 = 001000010010010000100..., in which 1 is in positions a(n) for n>=1. - _Clark Kimberling_, May 07 2017

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)

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

%t w1 = StringReplace[w, {"011" -> "0"}]

%t st = ToCharacterCode[w1] - 48 (* A286044 *)

%t Flatten[Position[st, 0]] (* A286045 *)

%t Flatten[Position[st, 1]] (* A003157 *)

%t (* _Clark Kimberling_, May 07 2017 *)

%Y Cf. A003156, A003158, A010060, A286044, A286045.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

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Last modified May 6 15:06 EDT 2021. Contains 343586 sequences. (Running on oeis4.)