A292200 Number of Sommerville symmetrical cyclic compositions (on symmetric necklaces) of n that are Carlitz (adjacent parts on the circle are distinct).

1, 1, 2, 2, 3, 4, 5, 7, 10, 11, 16, 23, 27, 37, 51, 65, 86, 117, 148, 204, 267, 351, 461, 626, 803, 1088, 1419, 1899, 2473, 3341, 4319, 5840, 7583, 10202, 13263, 17889, 23191, 31295, 40627, 54752, 71094, 95878, 124388, 167790, 217781, 293617, 381153, 513989, 667029, 899589

Offset: 1

Author

Petros Hadjicostas, Sep 11 2017

Keywords

• nonn

Comments

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.

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.

Examples

			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.)
		

Links

• Petros Hadjicostas, Cyclic, dihedral and symmetrical Carlitz compositions of a positive integer, Journal of Integer Sequences, 20 (2017), Article 17.8.5.
• Petros Hadjicostas, Generalized colored circular palindromic compositions, Moscow Journal of Combinatorics and Number Theory, 9(2) (2020), 173-186.
• P. Hadjicostas and L. Zhang, Sommerville's symmetrical cyclic compositions of a positive integer with parts avoiding multiples of an integer, Fibonacci Quarterly 55 (2017), 54-73.
• D. M. Y. Sommerville, On certain periodic properties of cyclic compositions of numbers, Proc. London Math. Soc. S2-7(1) (1909), 263-313.

Crossrefs

Cf. A119963, A291941.

Formula

a(n) = (A291941(n) + 1)/2.

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)).

Extensions

More terms from Altug Alkan, Sep 18 2017