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A292200 Number of Sommerville symmetrical cyclic compositions (on symmetric necklaces) of n that are Carlitz (adjacent parts on the circle are distinct). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

LINKS

Table of n, a(n) for n=1..50.

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.

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)).

EXAMPLE

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.)

CROSSREFS

Cf. A119963, A291941.

Sequence in context: A089047 A133498 A282949 * A249576 A097600 A136422

Adjacent sequences:  A292197 A292198 A292199 * A292201 A292202 A292203

KEYWORD

nonn

AUTHOR

Petros Hadjicostas, Sep 11 2017

EXTENSIONS

More terms from Altug Alkan, Sep 18 2017

STATUS

approved

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Last modified July 29 05:30 EDT 2021. Contains 346340 sequences. (Running on oeis4.)