cp's OEIS Frontend

A003144 Positions of letter a in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

%I A003144 M2399 #111 Jan 05 2025 19:51:33
%S A003144 1,3,5,7,8,10,12,14,16,18,20,21,23,25,27,29,31,32,34,36,38,40,42,44,
%T A003144 45,47,49,51,52,54,56,58,60,62,64,65,67,69,71,73,75,76,78,80,82,84,86,
%U A003144 88,89,91,93,95,97,99,101,102,104,106,108,110,112,113,115,117,119,121,123,125
%N A003144 Positions of letter a in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
%C A003144 From _Philippe Deléham_, Feb 27 2009: (Start)
%C A003144 A003144, A003145, A003146 may be defined as follows. Consider the morphism psi: a -> ab, b -> ac, c -> a. The image (or trajectory) of a under repeated application of this map is the infinite ternary tribonacci word a, b, a, c, a, b, a, a, b, a, c, a, b, a, b, a, c, ... (setting a = 1, b = 2, c = 3 gives A092782). The indices of a, b, c give respectively A003144, A003145, A003146. (End) [For the word with a -> 0, b -> 1, c -> 2 with offset 0 see A080843. - _Wolfdieter Lang_, Aug 10 2018]
%C A003144 The infinite word may also be defined as the limit S_oo where S_1 = a, S_n = psi(S_{n-1}). Or, by S_1 = a, S_2 = ab, S_3 = abac, and thereafter S_n = S_{n-1} S_{n-2} S_{n-3}. It is the unique word such that S_oo = psi(S_oo).
%C A003144 Also, indices of a in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - _Philippe Deléham_, Apr 16 2004
%C A003144 Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 0. [Duchene and Rigo, Remark 2.5] - _N. J. A. Sloane_, Nov 18 2016; corrected Mar 02 2019.
%D A003144 Eric Duchêne, Aviezri S. Fraenkel, Vladimir Gurvich, Nhan Bao Ho, Clark Kimberling, Urban Larsson, Wythoff Visions, Games of No Chance, Vol. 5;  MSRI Publications, Vol. 70 (2017), pages 101-153.
%D A003144 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003144 N. J. A. Sloane, <a href="/A003144/b003144.txt">Table of n, a(n) for n = 1..10609</a>
%H A003144 Elena Barcucci, Luc Belanger and Srecko Brlek, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/42-4/quartbarcucci04_2004.pdf">On tribonacci sequences</a>, Fib. Q., 42 (2004), 314-320.
%H A003144 L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/10-1/carlitz3-a.pdf">Fibonacci representations of higher order</a>, Fib. Quart., 10 (1972), 43-69. The present sequence is called a.
%H A003144 F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, <a href="https://www.combinatorics.org/ojs/index.php/eljc/article/view/v27i1p52/8039">Queens in exile: non-attacking queens on infinite chess boards</a>, Electronic J. Combin., 27:1 (2020), #P1.52.
%H A003144 Eric Duchêne and Michel Rigo, <a href="http://dx.doi.org/10.1051/ita:2007039">A morphic approach to combinatorial games: the Tribonacci case</a>, RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. doi:10.1051/ita:2007039. [Also available from <a href="http://archive.numdam.org/item/ITA_2008__42_2_375_0">Numdam archive</a>]
%H A003144 A. J. Hildebrand, Junxian Li, Xiaomin Li, and Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.
%H A003144 Wolfdieter Lang, <a href="https://arxiv.org/abs/1810.09787">The Tribonacci and ABC Representations of Numbers are Equivalent</a>, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
%H A003144 Jeffrey Shallit, <a href="https://arxiv.org/abs/2210.03996">Some Tribonacci conjectures</a>, arXiv:2210.03996 [math.CO], 2022.
%F A003144 It appears that a(n) is always either floor(n*t) or floor(n*t)+1 for all n, where t is the tribonacci constant A058265. See A275926. - _N. J. A. Sloane_, Oct 28 2016. This is true - see the Dekking et al. paper. - _N. J. A. Sloane_, Jul 22 2019
%p A003144 M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
%p A003144 for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
%p A003144 t0:=S[M]: l:=length(t0); t1:=[];
%p A003144 for i from 1 to l do if substring(t0,i..i) = `a` then t1:=[op(t1),i]; fi; od: t1; # _N. J. A. Sloane_, Nov 01 2006
%t A003144 A003144L = StringPosition[SubstitutionSystem[{"a" -> "ab", "b" -> "ac", "c" -> "a"}, "a", {#}][[1]], "a"][[All, 1]] &; A003144L[7] (* _JungHwan Min_, Dec 22 2016 *)
%Y A003144 Cf. A003145, A003146, A080843, A092782, A058265, A275926, A276793, A276796, A278039 (subtract 1 from each term, and use offset 0).
%Y A003144 First differences are A276788.
%Y A003144 For tribonacci representations of numbers see A278038.
%K A003144 nonn
%O A003144 1,2
%A A003144 _N. J. A. Sloane_
%E A003144 More terms from _Philippe Deléham_, Apr 16 2004
%E A003144 Entry revised by _N. J. A. Sloane_, Oct 13 2016