A276789 -id:A276789 - cp's OEIS frontend

A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

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

Offset: 1

N. J. A. Sloane

nonn

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

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).

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 b.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.
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.

Cf. A003144, A003146, A080843, A092782, A058265, A276799, A276800, A276794, A276797.
First differences give A276789. A278040 (subtract 1 from each term, and use offset 1).
For tribonacci representations of numbers see A278038.

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

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 *)

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

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

A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

1, 5, 8, 12, 14, 18, 21, 25, 29, 32, 36, 38, 42, 45, 49, 52, 56, 58, 62, 65, 69, 73, 76, 80, 82, 86, 89, 93, 95, 99, 102, 106, 110, 113, 117, 119, 123, 126, 130, 133, 137, 139, 143, 146, 150, 154, 157, 161, 163, 167, 170, 174, 178, 181, 185, 187, 191, 194, 198, 201, 205, 207, 211, 214, 218, 222, 225, 229, 231, 235

Offset: 0

N. J. A. Sloane, Nov 18 2016

This sequence gives the A(n) numbers of the W. Lang link. There the B(n) and C(n) numbers are A278039(n) and A278041(n), respectively. - Wolfdieter Lang, Dec 05 2018
Positions of letter b in the tribonacci word t generated by a->ab, b->ac, c->a, when given offset 0. - Michel Dekking, Apr 03 2019
This sequence gives the positions of the word ab in the tribonacci word t. This follows from the fact that the letter b is always preceded in t by the letter a, and the formula AA = B-1, where A := A003144, B := A003145, C := A003146. - Michel Dekking, Apr 09 2019

			The tribonacci representation of 7 is 1000 (see A278038), so a(7) has tribonacci representation 100001, which is 24+1 = 25, so a(7) = 25.

N. J. A. Sloane, Table of n, a(n) for n = 0..20000
L. Carlitz, R. Scoville and V. E. Hoggatt, Jr., Fibonacci representations of higher order, Fib. Quart., 10 (1972), 43-69.
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv:1810.09787v1 [math.NT], 2018.

Cf. A003145, A276789, A276796, A278038, A278039, A278041, A319198, A319968.

By analogy with the Wythoff compound sequences A003622 etc., the nine compounds of A003144, A003145, A003146 might be called the tribonacci compound sequences. They are A278040, A278041, and A319966-A319972.

a(n) = A003145(n+1) - 1.
a(n) = A003144(A003144(n)). - N. J. A. Sloane, Oct 05 2018
See Theorem 13 in the Carlitz, Scoville and Hoggatt paper. - Michel Dekking, Mar 20 2019
From Wolfdieter Lang, Dec 13 2018: (Start)
This sequence gives the indices k with A080843(k) = 1, ordered increasingly with offset 0.
a(n) = 1 + 4*n - A319198(n-1), n >= 0, with A319198(-1) = 0.
a(n) = A276796(C(n)) - 1, with C(n) = A278041(n).
For a proof see the W. Lang link, Proposition 5, and eq. (58).
a(n) - 1 = B1(n), where B1-numbers are B-numbers from A278039 followed by an A-number from A278040. See a comment and example in A319968.
a(n) - 1 = B(B(n)) = B(B(n) + 1) - 2, for n > = 0, where B = A278039.
(End)

A308199 The tribonacci representation of a(n) is obtained by appending 0,0 to the tribonacci representation of n (cf. A278038).

Original entry on oeis.org

0, 4, 7, 11, 13, 17, 20, 24, 28, 31, 35, 37, 41, 44, 48, 51, 55, 57, 61, 64, 68, 72, 75, 79, 81, 85, 88, 92, 94, 98, 101, 105, 109, 112, 116, 118, 122, 125, 129, 132, 136, 138, 142, 145, 149, 153, 156, 160, 162, 166, 169, 173, 177, 180, 184, 186, 190, 193, 197, 200, 204, 206, 210, 213, 217, 221, 224, 228

Offset: 0

N. J. A. Sloane, Jun 23 2019

From Michel Dekking, Oct 06 2019: (Start)
If w is a binary vector not containing 111, then w00 and w01 are also binary vectors not containing 111. So a(n) = A278040(n) - 1.
This sequence gives the positions of the word ab in the tribonacci word t, when t is given offset 0.
This sequence is the compound sequence A278039(A278039) of the three sequences A278039, A278040, A278041, which are the building blocks of the tribonacci world with offset 0. (End)

			u = abacabaabacaba.., then u(0)u(1) = ab, u(4)u(5) = ab, u(7)u(8) = ab, u(11)u(12) = ab.

Cf. A278038, A278039, A278040, A278041, A308200.

Essentially partial sums of A276789.

From Michel Dekking, Oct 06 2019: (Start)
a(n) = Sum_{k=1..n-1} d(k), where d is the tribonacci word on the alphabet {4,3,2}.
a(n) = A003144(A003144(n)) - 1. (End)

A276792 First differences of A003146.

Original entry on oeis.org

7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 6, 7, 4, 7, 6, 7, 7, 6, 7, 4, 7, 6, 7, 6, 7, 4, 7

Offset: 1

N. J. A. Sloane, Oct 27 2016

Rémy Sigrist, Table of n, a(n) for n = 1..10608
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See page 317.
F. Michel Dekking, Jeffrey Shallit, and N. J. A. Sloane, Queens in exile: non-attacking queens on infinite chess boards, Electronic J. Combin., 27:1 (2020), #P1.52.

Cf. A003144, A003145, A003146, A080843, A276788, A276789.

a(n) = A003146(n+1) - A003146(n), for n >= 1.
a(n) = A276788(n) + A276789(n) + 1, for n >= 1.
a(n+1) = 7 - t(n)*(t(n) + 1)/2, n >= 0, where t(n) = A080843(n). - Wolfdieter Lang, Dec 06 2018

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A003145 Positions of letter b in the tribonacci word abacabaabacababac... generated by a->ab, b->ac, c->a (cf. A092782).

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A278040 The tribonacci representation of a(n) is obtained by appending 0,1 to the tribonacci representation of n (cf. A278038).

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A308199 The tribonacci representation of a(n) is obtained by appending 0,0 to the tribonacci representation of n (cf. A278038).

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A276792 First differences of A003146.

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