A080843 -id:A080843 - cp's OEIS frontend

A319198 Partial sums of the infinite self-similar tribonacci word, written in the form A080843.

0, 1, 1, 3, 3, 4, 4, 4, 5, 5, 7, 7, 8, 8, 9, 9, 11, 11, 12, 12, 12, 13, 13, 15, 15, 16, 16, 18, 18, 19, 19, 19, 20, 20, 22, 22, 23, 23, 24, 24, 26, 26, 27, 27, 27, 28, 28, 30, 30, 31, 31, 31, 32, 32, 34, 34, 35, 35, 36, 36, 38, 38, 39, 39, 39, 40, 40, 42, 42, 43, 43

Offset: 0

Wolfdieter Lang, Oct 10 2018

This sequence produces a formula for the A-numbers A278040, specifying the positions (or indices) of 1's in A080843, namely A(n) = 4*n+1 - a(n-1), with a(-1) = 0.

Vincenzo Librandi, Table of n, a(n) for n = 0..10600
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.

Cf. A080843, A276797, A276798, A278039 (B-numbers), A278040 (A-numbers), A278041 (C-numbers).

a(n) = Sum_{j=0..n} A080843(n), n >= 0.

a(n) = z_A(n) + 2*z_C(n) = A276797(n+1) + 2*(A276798(n+1) - 1), where z_A(n) gives the number of A-numbers from A278040 not exceeding n, similarly for z_C(n) with the C-numbers from A278041. - Wolfdieter Lang, Dec 13 2018

A317950 First differences of ternary tribonacci word A080843.

Original entry on oeis.org

1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1, -1, 1, -1, 2, -2, 1, -1, 0, 1, -1, 2, -2, 1

Offset: 0

N. J. A. Sloane, Aug 17 2018

sign

Also first differences of A092782.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A080843, A092782.

A216190 Abelian complexity function of tribonacci word (A080843).

Original entry on oeis.org

3, 3, 4, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 5, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 5, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 3, 4, 4, 4

Offset: 1

Nathan Fox, Mar 11 2013

For all n, a(n) equals 3,4,5,6, or 7.
The values 3,4,5,6, and 7 are all obtained infinitely often.
The first 6 occurs when n=342.  The first 7 occurs when n=3914.

G. Richomme, K. Saari, L. Q. Zamboni, Balance and Abelian Complexity of the Tribonacci word, Adv. Appl. Math. 45 (2010) 212-231.

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Nathan Fox, Python code to generate sequence
Ondrej Turek, Abelian complexity and Abelian co-decomposition, arXiv 1201:2109, Jan. 11, 2012.
O. Turek, Abelian Complexity Function of the Tribonacci Word, J. Int. Seq. 18 (2015) # 15.3.4

Cf. A080843.

A345717 Orders of abelian cubes in the tribonacci word A080843.

Original entry on oeis.org

4, 6, 7, 11, 13, 17, 18, 20, 24, 26, 27, 30, 31, 33, 37, 38, 40, 41, 42, 43, 44, 48, 50, 51, 55, 57, 61, 62, 63, 64, 68, 70, 74, 75, 77, 79, 81, 85, 86, 87, 88, 92, 94, 95, 98, 99, 101, 105, 107, 108, 111, 112, 114, 116, 118, 119, 122, 123, 125, 129, 131, 132

Offset: 1

Jeffrey Shallit, Jun 24 2021

nonn

An abelian cube is a word of the form x x' x'', where x' and x'' are permutations of x, like the English word "deeded". The order of an abelian cube is the length of x.

			Here are the earliest-appearing abelian cubes of the first few orders:
n = 4:  2010.0102.0102
n = 6: 102010.010201.010201
n = 7: 0102010.0102010.1020100
n = 11: 02010010201.01020100102.01020100102

Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti, Additive word complexity and Walnut, arXiv:2410.02409 [math.CO], 2024. See p. 17.

Cf. A080843.

There is a deterministic finite automaton of 1169 states that takes n in its tribonacci representation as input and accepts if and only if there is an abelian cube of order n. It can be obtained with the Walnut theorem-prover.

A347752 Orders of additive cubes in the tribonacci word A080843.

Original entry on oeis.org

3, 4, 6, 7, 10, 11, 13, 14, 16, 17, 18, 20, 21, 23, 24, 26, 27, 30, 31, 33, 34, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 54, 55, 57, 58, 60, 61, 62, 63, 64, 65, 67, 68, 70, 71, 74, 75, 77, 78, 79, 81, 82, 84, 85, 86, 87, 88, 89, 91, 92, 94, 95, 97, 98, 99

Offset: 1

Jeffrey Shallit, Sep 18 2021

nonn

An additive cube is three consecutive blocks of the same length and same sum.

There is a tribonacci automaton of 4927 states recognizing the set of these orders (in tribonacci representation).

			The first few examples of additive cubes of different lengths in the tribonacci word are 020.101.020 (order 3), 2010.0102.0102 (order 4), and 102010.010201.010201 (order 6)

Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti, Additive word complexity and Walnut, arXiv:2410.02409 [math.CO], 2024. See p. 17.

Cf. A080843, A345717.

A275933 Decimal expansion of constant related to complexity of the tribonacci word (A080843).

Original entry on oeis.org

1, 0, 6, 0, 5, 2, 3, 8, 2, 9, 1, 0, 2, 6, 6, 3, 6, 1, 2, 0, 7, 9, 7, 2, 6, 9, 6, 3, 7, 5, 6, 3, 3, 5, 5, 7, 7, 4, 3, 2, 2, 9, 4, 2, 8, 3, 3, 5, 9, 4, 7, 4, 4, 6, 1, 0, 8, 1, 7, 8, 8, 3, 9, 9, 3, 8, 7, 4, 9, 4, 7, 0, 1, 4, 1, 0, 1, 8, 4, 7, 0, 1, 0, 1, 8, 5, 6, 2, 2, 0, 8, 7, 4, 3, 7, 0, 3, 9, 2, 9, 3, 3, 5, 0, 1, 8, 1

Offset: 2

N. J. A. Sloane, Sep 02 2016

The minimal polynomial of this constant is x^3 - 13*x^2 + 27*x - 17, and it is its unique real root. - Amiram Eldar, May 27 2023

			10.6052382910266361207972696375...

Nataliya Chekhova, Pascal Hubert, and Ali Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci, Journal de théorie des nombres de Bordeaux, 13.2 (2001): 371-394.

Cf. A080843, A058265.

RealDigits[x /. FindRoot[x^3 - 13*x^2 + 27*x - 17, {x, 10}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, May 27 2023 *)

Equals 9+(6*t-4)/(t^2+1), where t is the tribonacci constant A058265.

Equals (13 + (847 - 33*sqrt(33))^(1/3) + (11 * (77 + 3*sqrt(33)))^(1/3))/3. - Amiram Eldar, May 27 2023

More terms from Joerg Arndt, Sep 02 2016

A308525 Quasiperiods of the ternary tribonacci sequence (A080843).

Original entry on oeis.org

7, 13, 14, 24, 25, 26, 27, 44, 45, 46, 47, 48, 49, 50, 51, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176

Offset: 1

Jeffrey Shallit, Jun 11 2019

nonn

A word x is a quasiperiod of another (possibly infinite) word w if you can cover all of the symbols of w by translates of x.

H. Mousavi and J. Shallit, Mechanical Proofs of Properties of the Tribonacci Word, In: Manea F., Nowotka D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science, vol 9304. Springer, 2015, pp. 170-190.

Cf. A080843.

A308627 Numbers k such that the ternary tribonacci sequence (A080843) has a Lyndon factor of length k.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 11, 13, 16, 18, 20, 22, 24, 29, 31, 35, 37, 40, 42, 44, 53, 55, 57, 64, 66, 68, 77, 79, 81, 97, 99, 101, 110, 112, 121, 123, 125, 134, 136, 145, 147, 149, 178, 180, 189, 191, 193, 215, 217, 226, 228, 230, 246, 248, 250, 259, 261, 270

Offset: 1

Jeffrey Shallit, Jun 11 2019

nonn

A "factor" is a contiguous subblock. A factor is "Lyndon" if it is lexicographically least among all its cyclic shifts.

Hamoon Mousavi and Jeffrey Shallit, Mechanical Proofs of Properties of the Tribonacci Word, arXiv:1407.5841 [cs.FL], 2014.
H. Mousavi and J. Shallit, Mechanical Proofs of Properties of the Tribonacci Word, In: Manea F., Nowotka D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science, vol 9304. Springer, 2015, pp. 170-190.

Cf. A080843.

A345733 Numbers k such that there are two distinct abelian squares of order k in the tribonacci word A080843.

Original entry on oeis.org

15, 34, 59, 90, 96, 97, 102, 134, 137, 170, 171, 172, 178, 183, 215, 240, 252, 259, 262, 289, 321, 333, 364, 370, 371, 387, 389, 391, 402, 408, 411, 445, 457, 470, 482, 489, 516, 519, 538, 556, 557, 563, 594, 600, 601, 606, 638, 665, 674, 675, 676, 682, 687

Offset: 1

Jeffrey Shallit, Jun 25 2021

nonn

An abelian square is a word of the form x x' where x' is a permutation of x, like the English word "reappear". The order of an abelian square x x' is the length of x.

The tribonacci word has abelian squares of all orders. If we consider two abelian squares x x' and y y' to be the same if y is a permutation of x, then some orders have only 1 abelian square (up to this equivalence), while others have 2, and these are the only possibilities. There is a 463-state automaton that recognizes the tribonacci representation of those terms k in this sequence. All this can be proved with the Walnut theorem prover.

			For k = 15, the two distinct abelian squares are 100102010102010.010201001020101 and 020102010010201.010201001020102.

Cf. A080843.

A345901 a(n) is the largest k such that the tribonacci word A080843 contains a block of length k*n that is an abelian k-th power.

Original entry on oeis.org

2, 2, 2, 3, 2, 4, 5, 2, 2, 2, 4, 2, 8, 2, 2, 2, 4, 3, 2, 6, 2, 2, 2, 8, 2, 4, 3, 2, 2, 3, 6, 2, 3, 2, 2, 2, 10, 3, 2, 3, 3, 3, 3, 13, 2, 2, 2, 4, 2, 5, 4, 2, 2, 2, 5, 2, 6, 2, 2, 2, 6, 3, 3, 4, 2, 2, 2, 12, 2, 3, 2, 2, 2, 5, 5, 2, 3, 2, 3, 2, 21, 2, 2, 2, 4, 3

Offset: 1

Jeffrey Shallit, Jun 29 2021

nonn

An abelian k-th power consists of k blocks x_1 x_2 ... x_k such that each x_i is a permutation of x_1. For example, the English word "deeded" is an abelian 3rd power.

			For n=4, A080843 contains the block 2010.0102.0102, which is an abelian 3rd power. But no blocks of size 16 in A080843 are abelian 4th powers.

Cf. A080843.

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A319198 Partial sums of the infinite self-similar tribonacci word, written in the form A080843.

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A317950 First differences of ternary tribonacci word A080843.

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A216190 Abelian complexity function of tribonacci word (A080843).

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A345717 Orders of abelian cubes in the tribonacci word A080843.

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A347752 Orders of additive cubes in the tribonacci word A080843.

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A275933 Decimal expansion of constant related to complexity of the tribonacci word (A080843).

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A308525 Quasiperiods of the ternary tribonacci sequence (A080843).

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A308627 Numbers k such that the ternary tribonacci sequence (A080843) has a Lyndon factor of length k.

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A345733 Numbers k such that there are two distinct abelian squares of order k in the tribonacci word A080843.

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A345901 a(n) is the largest k such that the tribonacci word A080843 contains a block of length k*n that is an abelian k-th power.

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