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A003146 Positions of letter c in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
(Formerly M3407)
41
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_{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).

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

REFERENCES

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), 314-320.

L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69. The present sequence is called c.

Eric Duchêne and Michel Rigo, A morphic approach to combinatorial games: the Tribonacci case. RAIRO - Theoretical Informatics and Applications, 42, 2008, pp 375-393. 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

MAPLE

M:=17; S[1]:=`a`; S[2]:=`ab`; S[3]:=`abac`;

for n from 4 to M do S[n]:=cat(S[n-1], S[n-2], S[n-3]); 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).

Sequence in context: A248349 A246301 A160907 * A063237 A026381 A063556

Adjacent sequences:  A003143 A003144 A003145 * A003147 A003148 A003149

KEYWORD

nonn,changed

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

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Last modified January 19 07:43 EST 2019. Contains 319305 sequences. (Running on oeis4.)