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

%I A180322 #16 Jul 06 2018 17:10:46
%S A180322 1,1,3,6,15,21,49,72,126,195,341,486,819,1225,1845,2880,4335,6552,
%T A180322 9709,14850,21315,33077,47081,72360,102300,158067,220752,341334,
%U A180322 475107,732735,1015777,1566720,2161599,3333615,4587135,7062552,9699291,14922733,20444697
%N A180322 a(n) = AR(n) is the total number of aperiodic k-reverses of n.
%C A180322 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.
%C A180322 See sequence A180279 for the definition of an aperiodic k-reverse of n.
%C A180322 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.
%C A180322 For example a(6)=21 because there are 21 aperiodic k-reverses of n=6 for k=1,2,3,4,5, or 6.
%C A180322 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}.
%D A180322 John P. McSorley: Counting k-compositions with palindromic and related structures. Preprint, 2010.
%H A180322 Andrew Howroyd, <a href="/A180322/b180322.txt">Table of n, a(n) for n = 1..200</a>
%F A180322 a(n) = n * A056493(n) / 2. - _Andrew Howroyd_, Oct 07 2017
%t A180322 a[n_] := n*Sum[MoebiusMu[n/d]*If[OddQ[d], 2, 3]*2^Quotient[d-1, 2], {d, Divisors[n]}]/2;
%t A180322 Array[a, 40] (* _Jean-François Alcover_, Jul 06 2018, after _Andrew Howroyd_ *)
%o A180322 (PARI)
%o A180322 a(n) = n * sumdiv(n, d, moebius(n/d) * if(d%2,2,3) * 2^((d-1)\2)) / 2; \\ _Andrew Howroyd_, Oct 07 2017
%Y A180322 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.
%Y A180322 Row sums of A180279.
%K A180322 nonn
%O A180322 1,3
%A A180322 _John P. McSorley_, Aug 27 2010
%E A180322 Terms a(11) and beyond from _Andrew Howroyd_, Oct 07 2017