A179781 a(n) = AP(n) is the total number of aperiodic k-palindromes of n, 1 <= k <= n.

1, 1, 1, 2, 3, 5, 7, 12, 14, 27, 31, 54, 63, 119, 123, 240, 255, 490, 511, 990, 1015, 2015, 2047, 4020, 4092, 8127, 8176, 16254, 16383, 32607, 32767, 65280, 65503, 130815, 131061, 261576, 262143, 523775, 524223, 1047540, 1048575, 2096003, 2097151, 4192254

Offset: 1

Author

John P. McSorley, Jul 26 2010

Keywords

• nonn

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

Examples

			For a(7)=7, the achiral set partitions are 0000001, 0000011, 0000101, 0000111, 0001001, 0010011, and 0010101. - _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.

Crossrefs

Row sums of A179519.

A000048 (oriented), A000046 (unoriented), A308706 (chiral), A016116 (not primitive). - Robert A. Russell, Jun 19 2019

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[n/#] * 2^Floor[#/2]&];
Array[a, 44] (* Jean-François Alcover, Nov 04 2017 *)

PARI

a(n) = sumdiv(n, d, moebius(n/d) * 2^(d\2)); \\ Michel Marcus, Dec 09 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

Extensions

More terms from Michel Marcus, Dec 09 2014