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%I #30 Feb 22 2020 20:54:24
%S 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,
%T 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,
%U 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
%N Abelian complexity function of the 4-bonacci word (A254990).
%C For all n, a(n) either equals 4 or belongs to {6,7,...,16}; value 5 is never attained.
%C 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.
%C a(n)=6 only for n = 3,6,12.
%C Each value from the set {7,8,...,16} is attained infinitely often.
%H K. Brinda, <a href="http://brinda.cz/publications/kb_bach.pdf">Abelian complexity of infinite words</a>, bachelor thesis, Czech Technical University in Prague, 2011.
%H K. Brinda, <a href="http://brinda.cz/publications/vyzkumak.pdf">Abelian complexity of infinite words and Abelian return words</a>, Research project, Czech Technical University in Prague, 2012.
%H F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H O. Turek, <a href="http://arxiv.org/abs/1309.4810">Abelian complexity function of the Tribonacci word</a>, arXiv:1309.4810 [math.CO], 2013.
%H O. Turek, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Turek/turek3.html">Abelian complexity function of the Tribonacci word</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.4.
%e From _Wolfdieter Lang_, Mar 26 2015: (Start)
%e 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.
%e 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)
%Y Cf. A000078 (tetranacci numbers).
%Y Cf. A216190 (abelian complexity of tribonacci word), A254990 (4-bonacci word).
%K nonn
%O 1,1
%A _Ondrej Turek_, Feb 12 2015