

A003144


Positions of letter a in the tribonacci word abacabaabacababac... generated by a>ab, b>ac, c>a (cf. A092782).
(Formerly M2399)


49



1, 3, 5, 7, 8, 10, 12, 14, 16, 18, 20, 21, 23, 25, 27, 29, 31, 32, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 76, 78, 80, 82, 84, 86, 88, 89, 91, 93, 95, 97, 99, 101, 102, 104, 106, 108, 110, 112, 113, 115, 117, 119, 121, 123, 125
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OFFSET

1,2


COMMENTS

Comment from Philippe Deléham, Feb 27 2009: (Start)
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]
The infinite word may also be defined as the limit S_oo where S_1 = a, S_n = psi(S_{n1}). Or, by S_1 = a, S_2 = ab, S_3 = abac, and thereafter S_n = S_{n1} S_{n2} S_{n3}. It is the unique word such that S_oo = psi(S_oo).
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
Theorem: A number m is in this sequence iff the tribonacci representation of m1 ends with 0. [Duchene and Rigo, Remark 2.5]  N. J. A. Sloane, Nov 18 2016; corrected Mar 02 2019.


REFERENCES

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 101153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

N. J. A. Sloane, Table of n, a(n) for n = 1..10609
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314320.
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 4369. The present sequence is called a.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: nonattacking queens on infinite chess boards, arXiv:1907.09120, July 2019
Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case, RAIRO  Theoretical Informatics and Applications, 42, 2008, pp 375393. doi:10.1051/ita:2007039. [Also available from Numdam archive]
A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.


FORMULA

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


MAPLE

M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;
for n from 4 to M do S[n]:=cat(S[n1], S[n2], S[n3]); od:
t0:=S[M]: l:=length(t0); t1:=[];
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


MATHEMATICA

A003144L = StringPosition[SubstitutionSystem[{"a" > "ab", "b" > "ac", "c" > "a"}, "a", {#}][[1]], "a"][[All, 1]] &; A003144L[7] (* JungHwan Min, Dec 22 2016 *)


CROSSREFS

Cf. A003145, A003146, A080843, A092782, A058265, A275926, A276793, A276796, A278039 (subtract 1 from each term, and use offset 0).
First differences are A276788.
For tribonacci representations of numbers see A278038.
Sequence in context: A059565 A187841 A253301 * A247357 A191251 A285594
Adjacent sequences: A003141 A003142 A003143 * A003145 A003146 A003147


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Philippe Deléham, Apr 16 2004
Entry revised by N. J. A. Sloane, Oct 13 2016


STATUS

approved



