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A003144 Positions of letter a in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
(Formerly M2399)
42
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 (list; graph; refs; listen; history; text; internal format)
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_{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 a in the sequence closed under a -> abac, b -> aba, c -> ab; starting with a(1) = a. - Philippe Deléham, Apr 16 2004

The tribonacci representation of a(n) is obtained by appending a 1 to the tribonacci representation of n-1 (cf. A278038). For example, a(8) = 14, which has tribonacci representation 10001, and is indeed obtained by appending a 1 to the tribonacci representation of 7, which is 1000. - N. J. A. Sloane, Nov 18 2016

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 a.

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 archive]

A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [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

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) = `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.

Sequence in context: A059565 A187841 A253301 * A247357 A191251 A285594

Adjacent sequences:  A003141 A003142 A003143 * A003145 A003146 A003147

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 October 21 02:41 EDT 2018. Contains 316405 sequences. (Running on oeis4.)