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

Author

Ondrej Turek, Feb 12 2015

Keywords

• nonn

Comments

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.

Examples

			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)
		

Links

• 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.

Crossrefs

Cf. A000078 (tetranacci numbers).

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