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User: Ondrej Turek

Ondrej Turek has authored 2 sequences.

A255014 Abelian complexity function of the 4-bonacci word (A254990).

4, 4, 6, 4, 7, 6, 7, 4, 7, 7, 8, 6, 8, 7, 7, 4, 7, 7, 8, 7, 8, 8, 7, 7, 8, 8, 7, 8, 7, 7, 4, 7, 7, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 7, 7, 7, 7, 8, 8, 8, 8, 7, 8, 8, 8, 7, 8, 7, 7, 4, 7, 8, 9, 7, 8, 9, 9, 7, 8, 10, 10, 8, 8, 8, 8, 7, 9, 10, 9, 8, 9, 9, 8, 8, 9, 10, 7, 8, 7, 8, 7, 8, 9, 9, 8, 8, 8, 8, 8, 7

Offset: 1

Ondrej Turek, Feb 12 2015

nonn

For all n, a(n) either equals 4 or belongs to {6,7,...,16}; value 5 is never attained.
a(n)=4 if and only if n = T(k)+T(k-4)+T(k-8)+T(k-12)+...+T(4+(k mod 4)) for a certain k>=4, where T(i) are tetranacci numbers A000078.
a(n)=6 only for n = 3,6,12.
Each value from the set {7,8,...,16} is attained infinitely often.

			From _Wolfdieter Lang_, Mar 26 2015: (Start)
a(1) = 4 because the one letter factor words of A254990 are 0, 1, 2, 3 with the set of occurrence tuples (Parikh vectors) {(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1)} of cardinality 4. See the Turek links.
a(2) = 4 because the set of occurrence tuples for the two letter factors 00, 01, 10, 02, 20, 03, 30 of A254990 is {(2, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 0, 1)} of cardinality 4. (End)

K. Brinda, Abelian complexity of infinite words, bachelor thesis, Czech Technical University in Prague, 2011.
K. Brinda, Abelian complexity of infinite words and Abelian return words, Research project, Czech Technical University in Prague, 2012.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
O. Turek, Abelian complexity function of the Tribonacci word, arXiv:1309.4810 [math.CO], 2013.
O. Turek, Abelian complexity function of the Tribonacci word, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.4.

Cf. A000078 (tetranacci numbers).

Cf. A216190 (abelian complexity of tribonacci word), A254990 (4-bonacci word).

A254990 4-bonacci word. Fixed point of morphism 0->01, 1->02, 2->03, 3->0.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3

Offset: 0

Ondrej Turek, Feb 11 2015

Special case of k-bonacci word for k = 4 (see crossrefs).
The lengths of iterations S(i) are Tetranacci numbers (A000078).
Set S(0) = 0; S(1) = 0,1; S(2) = 0,1,0,2; S(3) = 0,1,0,2,0,1,0,3; for n >= 4: S(n) = S(n-1) S(n-2) S(n-3) S(n-4). The sequence is the limit S(infinity).

			The iterates are:
0
01
0102
01020103
010201030102010
01020103010201001020103010201
01020103010201001020103010201010201030102010010201030102
...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Elena Barcucci, Luc Belanger and Srecko Brlek, On tribonacci sequences, Fib. Q., 42 (2004), 314-320. See Section 4.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
O. Turek, Abelian Complexity Function of the Tribonacci Word, J. Int. Seq. 18 (2015) # 15.3.4

Cf. A000078 (lengths of iterations).
Cf. A003849 (k=2, Fibonacci word), A080843 (k=3, Tribonacci word).
Cf. A316837, A316838, A316839, A316840.

Nest[Flatten[#/.{0->{0,1},1->{0,2},2->{0,3},3->0}]&,0,7] (* Harvey P. Dale, Mar 26 2015 *)

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User: Ondrej Turek

A255014 Abelian complexity function of the 4-bonacci word (A254990).

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