

A180322


a(n) = AR(n) is the total number of aperiodic kreverses of n.


4



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
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OFFSET

1,3


COMMENTS

The nth term of this sequence a(n) = AR(n) gives the total number of aperiodic kreverses of n for k=1,2,...,n. It is the sum of the nth row of the 'AR(n,k)' triangle from sequence A180279.
See sequence A180279 for the definition of an aperiodic kreverse of n.
Briefly, a kreverse of n is a kcomposition of n whose reverse is cyclically equivalent to itself, and an aperiodic kreverse of n is a kreverse of n which is also aperiodic.
For example a(6)=21 because there are 21 aperiodic kreverses 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 kcompositions 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[d1, 2], {d, Divisors[n]}]/2;
Array[a, 40] (* JeanFrançois Alcover, Jul 06 2018, after Andrew Howroyd *)


PROG

(PARI)
a(n) = n * sumdiv(n, d, moebius(n/d) * if(d%2, 2, 3) * 2^((d1)\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



