OFFSET
1,4
COMMENTS
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 a smaller composition.
A k-palindrome of n is a k-composition of n which is a palindrome.
This sequence is AP(n), the total number of aperiodic k-palindromes of n, 1 <= k <= n.
For example AP(6)=5 because the number n=6
has 1 aperiodic 1-palindrome, namely 6 itself;
has 1 aperiodic 3-palindrome, namely 141;
has 2 aperiodic 4-palindromes, namely 2112 and 1221;
has 1 aperiodic 5-palindrome, namely 11211.
This gives a total of 1+1+2+1=5 aperiodic palindromes of 6.
Number of achiral set partitions of a primitive cycle of n elements having up to two different elements. - Robert A. Russell, Jun 19 2019
REFERENCES
John P. McSorley, Counting k-compositions of n with palindromic and related structures. Preprint, 2010.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..200
Hunki Baek, Sejeong Bang, Dongseok Kim, Jaeun Lee, A bijection between aperiodic palindromes and connected circulant graphs, arXiv:1412.2426 [math.CO], 2014.
FORMULA
a(n) = Sum_{d | n} moebius(n/d)*2^(floor(d/2)) (see Baek et al. page 9). - Michel Marcus, Dec 09 2014
a(n) = 2*A000046(n) - A000048(n) = A000048(n) - 2*A308706(n) = A000046(n) - A308706(n). - Robert A. Russell, Jun 19 2019
A016116(n) = Sum_{d|n} a(d). - Robert A. Russell, Jun 19 2019
G.f.: Sum_{k>=1} mu(k)*x^k*(1 + 2*x^k)/(1 - x^(2*k)). - Andrew Howroyd, Sep 27 2019
EXAMPLE
For a(7)=7, the achiral set partitions are 0000001, 0000011, 0000101, 0000111, 0001001, 0010011, and 0010101. - Robert A. Russell, Jun 19 2019
MATHEMATICA
a[n_] := DivisorSum[n, MoebiusMu[n/#] * 2^Floor[#/2]&];
Array[a, 44] (* Jean-François Alcover, Nov 04 2017 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(n/d) * 2^(d\2)); \\ Michel Marcus, Dec 09 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
John P. McSorley, Jul 26 2010
EXTENSIONS
More terms from Michel Marcus, Dec 09 2014
STATUS
approved