%I #58 Sep 29 2023 05:01:43
%S 1,1,2,2,3,4,5,7,10,11,16,23,27,37,51,65,86,117,148,204,267,351,461,
%T 626,803,1088,1419,1899,2473,3341,4319,5840,7583,10202,13263,17889,
%U 23191,31295,40627,54752,71094,95878,124388,167790,217781,293617,381153,513989,667029,899589
%N Number of Sommerville symmetrical cyclic compositions (on symmetric necklaces) of n that are Carlitz (adjacent parts on the circle are distinct).
%C We consider cyclic compositions (necklaces) as equivalence classes of compositions that can be obtained from each other by a cyclic shift. A cyclic composition is called Sommerville symmetrical (on a symmetric necklace) if its equivalence class contains at least one palindromic composition (type I) or a composition that becomes a palindromic composition if we remove the first part (type II). A composition with only one part is a palindromic composition of both types.
%C The equivalence class of each Sommerville symmetrical cyclic composition that is Carlitz contains exactly two type II palindromic Carlitz compositions (except in the case of a composition with only one part). For example, when n = 8, the equivalence class {(1,2,3,2), (2,3,2,1), (3,2,1,2), (2,1,2,3)} represents a Sommerville symmetrical cyclic composition of n = 8 that is Carlitz, but only two of the compositions in the set, i.e., (1,2,3,2) and (3,2,1,2), are type II palindromic.
%H Petros 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.
%H Petros Hadjicostas, <a href="https://doi.org/10.2140/moscow.2020.9.173">Generalized colored circular palindromic compositions</a>, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
%H P. Hadjicostas and L. Zhang, <a href="https://www.fq.math.ca/Papers1/55-1/HadjicostasZhang10252016.pdf">Sommerville's symmetrical cyclic compositions of a positive integer with parts avoiding multiples of an integer</a>, Fibonacci Quarterly 55 (2017), 54-73.
%H D. M. Y. Sommerville, <a href="https://doi.org/10.1112/plms/s2-7.1.263">On certain periodic properties of cyclic compositions of numbers</a>, Proc. London Math. Soc. S2-7(1) (1909), 263-313.
%F a(n) = (A291941(n) + 1)/2.
%F G.f.: x/(1 - x) - A(x)/2 + B(x)^2/(2*(1 - A(x)), where A(x) = Sum_{n >= 1} x^(2*n)/(1 + x^(2*n)) and B(x) = Sum_{n >= 1} x^n/(1 + x^(2*n)).
%e For n = 7, there are exactly a(7) = 5 Sommerville symmetrical cyclic compositions (symmetric necklaces) of 7 that are Carlitz: 7, 1+6, 2+5, 3+4, 2+1+3+1. (Note that 1+6 is the same as 6+1, 3+1+2+1 is the same as 2+1+3+1, and so on, because in each case one composition can be obtained from the other by a cyclic shift.)
%Y Cf. A119963, A291941.
%K nonn
%O 1,3
%A _Petros Hadjicostas_, Sep 11 2017
%E More terms from _Altug Alkan_, Sep 18 2017