

A003146


Positions of letter c in the tribonacci word abacabaabacababac... generated by a>ab, b>ac, c>a (cf. A092782).
(Formerly M3407)


50



4, 11, 17, 24, 28, 35, 41, 48, 55, 61, 68, 72, 79, 85, 92, 98, 105, 109, 116, 122, 129, 136, 142, 149, 153, 160, 166, 173, 177, 184, 190, 197, 204, 210, 217, 221, 228, 234, 241, 247, 254, 258, 265, 271, 278, 285, 291, 298, 302, 309, 315, 322, 329, 335, 342, 346, 353, 359
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Comment from Philippe Deléham, Feb 27 2009: 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.
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 c 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 11. [Duchene and Rigo, Remark 2.5]  N. J. A. Sloane, 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 c.
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]
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) = floor(n*t^3) + eps for all n, where t is the tribonacci constant A058265 and eps is 0, 1, 2, or 3. See A277721.  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) = `c` then t1:=[op(t1), i]; fi; od:
# N. J. A. Sloane, Nov 01 2006


MATHEMATICA

StringPosition[SubstitutionSystem[{"a" > "ab", "b" > "ac", "c" > "a"}, "c", {#}][[1]], "c"][[All, 1]] &@ 11 (* Michael De Vlieger, Mar 30 2017, Version 10.2, after JungHwan Min at A003144 *)


CROSSREFS

Cf. A003145, A003144, A080843, A092782, A058265, A276791, A276798, A276801, A277721.
First differences are A276792. A278041 (subtract 1 from each term, and use offset 0).
For tribonacci representations of numbers see A278038.
Sequence in context: A248349 A246301 A160907 * A063237 A026381 A063556
Adjacent sequences: A003143 A003144 A003145 * A003147 A003148 A003149


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



