cp's OEIS Frontend

A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

%I A003145 M1571 #77 Jan 05 2025 19:51:33
%S A003145 2,6,9,13,15,19,22,26,30,33,37,39,43,46,50,53,57,59,63,66,70,74,77,81,
%T A003145 83,87,90,94,96,100,103,107,111,114,118,120,124,127,131,134,138,140,
%U A003145 144,147,151,155,158,162,164,168,171,175,179,182,186,188,192,195,199,202,206,208
%N A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
%C A003145 A003144, A003145, A003146 may be defined as follows. Consider the map psi: a -> ab, b -> ac, c -> a. The image (or trajectory) of a under repeated application of this map is the infinite 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. - _Philippe Deléham_, Feb 27 2009
%C A003145 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 A003145 Also indices of b in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - _Philippe Deléham_, Apr 16 2004
%C A003145 Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 01. [Duchene and Rigo, Remark 2.5] - _N. J. A. Sloane_, Mar 02 2019
%D A003145 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 A003145 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A003145 N. J. A. Sloane, <a href="/A003145/b003145.txt">Table of n, a(n) for n = 1..10609</a>
%H A003145 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 A003145 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 b.
%H A003145 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 A003145 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</a>]
%H A003145 A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, <a href="https://arxiv.org/abs/1809.08690">Almost Beatty Partitions</a>, arXiv:1809.08690 [math.NT], 2018.
%H A003145 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.
%F A003145 It appears that a(n) = floor(n*t^2) + eps for all n, where t is the tribonacci constant A058265 and eps is 0, 1, or 2. See A276799. - _N. J. A. Sloane_, Oct 28 2016. This is true - see the Dekking et al. paper. - _N. J. A. Sloane_, Jul 22 2019
%p A003145 M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
%p A003145 for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); od:
%p A003145 t0:=S[M]: l:=length(t0); t1:=[];
%p A003145 for i from 1 to l do if substring(t0,i..i) = `b` then t1:=[op(t1),i]; fi; od: # _N. J. A. Sloane_
%t A003145 StringPosition[SubstitutionSystem[{"a" -> "ab", "b" -> "ac", "c" -> "a"}, "b", {#}][[1]], "b"][[All, 1]] &@9 (* _Michael De Vlieger_, Mar 30 2017, Version 10.2, after _JungHwan Min_ at A003144 *)
%Y A003145 Cf. A003144, A003146, A080843, A092782, A058265, A276799, A276800, A276794, A276797.
%Y A003145 First differences give A276789. A278040 (subtract 1 from each term, and use offset 1).
%Y A003145 For tribonacci representations of numbers see A278038.
%K A003145 nonn
%O A003145 1,1
%A A003145 _N. J. A. Sloane_
%E A003145 More terms from _Philippe Deléham_, Apr 16 2004
%E A003145 Corrected by _T. D. Noe_ and _N. J. A. Sloane_, Nov 01 2006
%E A003145 Entry revised by _N. J. A. Sloane_, Oct 13 2016