A180322 a(n) = AR(n) is the total number of aperiodic k-reverses of n.

1, 1, 3, 6, 15, 21, 49, 72, 126, 195, 341, 486, 819, 1225, 1845, 2880, 4335, 6552, 9709, 14850, 21315, 33077, 47081, 72360, 102300, 158067, 220752, 341334, 475107, 732735, 1015777, 1566720, 2161599, 3333615, 4587135, 7062552, 9699291, 14922733, 20444697

Offset

1,3

Comments

The n-th term of this sequence a(n) = AR(n) gives the total number of aperiodic k-reverses of n for k=1,2,...,n. It is the sum of the n-th row of the 'AR(n,k)' triangle from sequence A180279.

See sequence A180279 for the definition of an aperiodic k-reverse of n.

Briefly, a k-reverse of n is a k-composition of n whose reverse is cyclically equivalent to itself, and an aperiodic k-reverse of n is a k-reverse of n which is also aperiodic.

For example a(6)=21 because there are 21 aperiodic k-reverses of n=6 for k=1,2,3,4,5, or 6.

They are, in cyclically equivalent classes: {6}, {15,51}, {24,42}, {114,411,141}, {1113,3111,1311,1131}, {1122,2112,2211,1221},{11112,21111,12111,11211,11121}.

References

John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(n) = n * A056493(n) / 2. - Andrew Howroyd, Oct 07 2017

Mathematica

a[n_] := n*Sum[MoebiusMu[n/d]*If[OddQ[d], 2, 3]*2^Quotient[d-1, 2], {d, Divisors[n]}]/2;
Array[a, 40] (* Jean-François Alcover, Jul 06 2018, after Andrew Howroyd *)

Prog

(PARI)
a(n) = n * sumdiv(n, d, moebius(n/d) * if(d%2, 2, 3) * 2^((d-1)\2)) / 2; \\ Andrew Howroyd, Oct 07 2017

Crossrefs

If we ask for the number of cyclically equivalent classes we get sequence A056493 (except for the first term). For example, the 6th term of A056493 is 7, corresponding to the 7 classes in the example above.

Row sums of A180279.

Sequence in context: A093799 A087359 A253651 * A244164 A129602 A044888

Adjacent sequences: A180319 A180320 A180321 * A180323 A180324 A180325

Keyword

nonn

Author

John P. McSorley, Aug 27 2010

Extensions

Terms a(11) and beyond from Andrew Howroyd, Oct 07 2017

Status

approved