

A003145


Positions of letter b in the tribonacci word abacabaabacababac... generated by a>ab, b>ac, c>a (cf. A092782).
(Formerly M1571)


41



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
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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_{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 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), 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 b.
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^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[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) = `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. A278040 (subtract 1 from each term, and use offset 1).
Sequence in context: A236760 A086562 A083789 * A184621 A184821 A292659
Adjacent sequences: A003142 A003143 A003144 * A003146 A003147 A003148


KEYWORD

nonn,changed


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



