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A003145
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Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).
(Formerly M1571)
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57
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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
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1,1
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COMMENTS
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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
Theorem: A number m is in this sequence iff the tribonacci representation of m-1 ends with 01. [Duchene and Rigo, Remark 2.5] - N. J. A. Sloane, Mar 02 2019
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REFERENCES
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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 101-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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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. This is true - see the Dekking et al. paper. - N. J. A. Sloane, Jul 22 2019
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MAPLE
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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
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MATHEMATICA
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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 *)
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CROSSREFS
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First differences give A276789. A278040 (subtract 1 from each term, and use offset 1).
For tribonacci representations of numbers see A278038.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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