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A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. A k-composition is aperiodic (primitive) if its period is k, i.e., if it is not the concatenation of at least two smaller compositions.

A palindrome is a word which is the same when written backwards.

A k-double-palindrome of n is a k-composition of n which is the concatenation of two palindromes, PP'=P|P', where both |P|, |P'|>=1.

For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes. It is also aperiodic, and so it is an aperiodic 7-double-palindrome of 17. The 4-double-palindrome of 8 1313=131|3 is not aperiodic, so it is not an aperiodic 4-double-palindrome of 8.

Let ADP(n,k) denote the number of aperiodic k-double-palindromes of n.

This sequence is the 'ADP(n,k)' triangle read by rows.

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.

Two k-compositions are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.

The reverse of a k-composition is the k-composition obtained by writing its parts in reverse.

For example the reverse of 123 is 321.

A k-reverse of n is a k-composition of n which is cyclically equivalent to its reverse.

For example 114 is a 3-reverse of 6 since its set of cyclic equivalents {114,411,141} contains its reverse 411. But 123 is not a 3-reverse of 6 since its set of cyclic equivalents {123,312,231} does not contain its reverse 321.

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n. A k-composition is aperiodic (primitive) if its period is k, i.e., if it is not the concatenation of at least two smaller compositions. A palindrome is a word which is the same when written backwards.

A k-double-palindrome of n is a k-composition of n which is the concatenation of two palindromes, PP'=P|P', where both |P|, |P'|>=1.

For example 1123532=11|23532 is a 7-double-palindrome of 17 since both 11 and 23532 are palindromes. It is also aperiodic, and so it is an aperiodic 7-double-palindrome of 17. See sequence A181111.

Two k-compositions of n are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.

Let ADPE(n,k) denote the number of aperiodic k-double-palindromes of n up to cyclic equivalence, i.e., the number of cyclic equivalence classes containing at least one aperiodic k-double-palindrome.

This sequence is the `ADPE(n,k)' triangle read by rows.

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.

A k-composition of n is an ordered collection of k positive integers (parts) which sum to n.

A k-composition is aperiodic (primitive) if its period is k, or if it is not the concatenation of at least two smaller compositions.

Two k-compositions are cyclically equivalent if one can be obtained from the other by a cyclic permutation of its parts.

The reverse of a k-composition is the k-composition obtained by writing its parts in reverse. For example, the reverse of 123 is 321.

A k-reverse of n is a k-composition of n which is cyclically equivalent to its reverse. And an aperiodic k-reverse of n is a k-reverse of n which is aperiodic.

For example, 114 is an aperiodic 3-reverse of 6 since it is aperiodic and its set of cyclic equivalents {114,411,141} contains its reverse 411.

But 123 is not an aperiodic 3-reverse of 6 since, even though it is aperiodic, its set of cyclic equivalents {123,312,231} does not contain its reverse 321.

Let AR(n,k) denote the number of aperiodic k-reverses of n, then sequence A180279 is the 'AR(n,k)' triangle read by rows.

For the above sequence we count the aperiodic k-reverses of n up to cyclic equivalence, ARE(n,k), in other words, the number of equivalence classes under cyclic permutation which contain at least one aperiodic k-reverse of n.

Bracelets that have no reflection symmetry are also known as chiral bracelets.

Here, for n >= 6, a(n) is also the number of dihedral compositions of n with 6 parts that have no reflection symmetry. Taking the MacMahon conjugates of these dihedral compositions, we see that a(n) is also the number of dihedral compositions of n into n-6 parts that have no reflection symmetry.

A cyclic composition b_1 + b_2 + ... + b_k of n into k parts is an equivalent class of (linear) compositions of n into k parts (placed on a circle) such that two such (linear) compositions are equivalent iff one can be obtained from the other by a rotation. Such compositions were first studied extensively by Sommerville (1909).

A dihedral composition b_1 + b_2 + ... + b_k of n into k parts is an equivalent class of (linear) compositions of n into k parts (placed on a circle) such that two such (linear) compositions are equivalent iff one can be obtained from the other by a rotation or a reversal of order. Such compositions were studied, for example, by Knopfmacher and Robbins (2013).

Given a bracelet of length n with k white beads and n-k black beads, we may get the corresponding dihedral composition using MacMahon's correspondence: start with a white bead and count that bead and the black beads that follow (in one direction), and call that b_1; then start with the next white bead and count that one and the black beads that follow, and call that b_2; repeat this process until you reach the k-th white bead and count that one and the black beads that follow, and call that b_k. The corresponding dihedral composition is b_1 + b_2 + ... + b_k.

If in the previous paragraph (given a bracelet of length n with k white beads and n-k black beads), we replace the white beads with black beads and the black beads with white beads, we get a dihedral composition of n into n-k parts: c_1 + c_2 + ... + c_{n-k}. These two dihedral compositions (which correspond to the same bracelet) are called "conjugate" compositions. See p. 273 in Sommerville (1909) for an explanation of "conjugate" compositions in the context of cyclic compositions.

Symmetric cyclic compositions of a positive integer n were first studied by Sommerville (1909, pp. 301-304). It can be proved that the study of necklaces with reflection symmetry using beads of two colors is equivalent to the study of symmetric cyclic compositions of a positive integer. Clearly all the necklaces with reflection symmetry are all the bracelets (turnover necklaces) with reflection symmetry. See also the comments for sequences A119963, A292200, and A295925.