%I #40 Nov 02 2017 15:11:41
%S 1,1,2,2,3,5,6,9,14,20,29,48,69,110,175,278,441,725,1168,1928,3170,
%T 5253,8710,14563,24308,40798,68520,115433,194611,328938,556336,942659,
%U 1598539,2714379,4612681,7847082,13358850,22762311,38810771,66223599,113067441,193172332
%N Number of dihedral Carlitz compositions of n.
%C A cyclic Carlitz composition is a composition of length greater than one where adjacent parts, including the first and the last ones, are distinct. A composition of length one is also considered cyclic and Carlitz. Assume two cyclic Carlitz compositions are considered equivalent iff one can be obtained from the other by a rotation or reversal of order. Each equivalence class obtained is called a dihedral Carlitz composition of n.
%H P. Hadjicostas, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Hadjicostas/hadji5.html">Cyclic, dihedral and symmetrical Carlitz compositions of a positive integer</a>, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
%F a(n) = (A106369(n) + A292200(n))/2.
%F a(n) = (2*A106369(n) + A291941(n) + 1)/4.
%F G.f.: (g.f. of A106369 + g.f. of A292200)/2.
%e a(6) = 5 because n = 6 has the following dihedral Carlitz compositions: 6, 1+5, 2+4, 1+2+3, 1+2+1+2. (For example, the equivalence class for the dihedral Carlitz composition 1+2+3 is {(1,2,3),(2,3,1), (3,1,2), (3,2,1),(2,1,3),(1,3,2)}.)
%Y Cf. A106369, A291941, A292200.
%K nonn
%O 1,3
%A _Petros Hadjicostas_, Oct 10 2017