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A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
(Formerly M1571)
27
2, 6, 9, 13, 15, 19, 22, 26, 30, 33, 37, 39, 43, 46, 50, 53, 57, 59, 63, 66, 70, 74, 77, 81, 83, 87, 90, 94, 96, 100, 103, 107, 111, 114, 118, 120, 124, 127, 131, 134, 138, 140, 144, 147, 151, 155, 158, 162, 164, 168, 171, 175, 179, 182, 186, 188, 192, 195, 199, 202, 206, 208 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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 b 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.

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 http://archive.numdam.org/ARCHIVE/ITA/ITA_2008__42_2/ITA_2008__42_2_375_0/ITA_2008__42_2_375_0.pdf]

FORMULA

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

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) = `b` then t1:=[op(t1), i]; fi; od: # N. J. A. Sloane

MATHEMATICA

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 *)

CROSSREFS

Cf. A003144, A003146, A080843, A092782, A058265, A276799, A276800, A276794, A276797.

First differences give A276789.

Sequence in context: A236760 A086562 A083789 * A184621 A184821 A292659

Adjacent sequences:  A003142 A003143 A003144 * A003146 A003147 A003148

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Philippe Deléham, Apr 16 2004

Corrected by T. D. Noe and N. J. A. Sloane, Nov 01 2006

Entry revised by N. J. A. Sloane, Oct 13 2016

STATUS

approved

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Last modified February 22 04:10 EST 2018. Contains 299428 sequences. (Running on oeis4.)