A180249 a(n) is the total number of k-reverses of n.

1, 2, 4, 8, 16, 26, 50, 80, 130, 212, 342, 518, 820, 1276, 1864, 2960, 4336, 6704, 9710, 15068, 21368, 33420, 47082, 72950, 102316, 158888, 220882, 342616, 475108, 734816, 1015778, 1569680, 2161944, 3337952, 4587200, 7069748, 9699292, 14932444, 20445520

Offset: 1

Author

John P. McSorley, Aug 19 2010

Keywords

• nonn

Comments

See sequence A180171 for the definition of a k-reverse of n.

Briefly, a k-reverse of n is a k-composition of n whose reverse is cyclically equivalent to itself.

This sequence is the total number of k-reverses of n for k=1,2,...,n.

It is the row sums of the 'R(n,k)' triangle from sequence A180171.

For example a(6)=26 because there are 26 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},{33},{114,411,141},{222} {1113,3111,1311,1131}, {1122,2112,2211,1221}, {1212,2121}, {11112,21111,12111,11211,11121}, {111111}.

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

Crossrefs

If we ask for the number of cyclically equivalent classes we get sequence A052955.

For example the 6th term of A052955 is 11, corresponding to the 11 classes in the example above.

Row sums of A180171.

Cf. A056493, A180322.

Programs

Mathematica

f[n_Integer] := Block[{c = 0, k = 1, ip = IntegerPartitions@ n, lmt = 1 + PartitionsP@ n, ipk}, While[k < lmt, c += g[ ip[[k]]]; k++ ]; c]; g[lst_List] := Block[{c = 0, len = Length@ lst, per = Permutations@ lst}, While[ Length@ per > 0, rl = Union[ RotateLeft[ per[[1]], # ] & /@ Range@ len]; If[ MemberQ[rl, Reverse@ per[[1]]], c += Length@ rl]; per = Complement[ per, rl]]; c]; Array[f, 24] (* Robert G. Wilson v, Aug 25 2010 *)
b[n_] := Sum[MoebiusMu[n/d] * If[OddQ[d], 2, 3] * 2^Quotient[d-1, 2], {d, Divisors[n]}]; a[n_] := Sum[d*b[d], {d, Divisors[n]}] / 2; Array[a, 39] (* Jean-François Alcover, Nov 04 2017, after Andrew Howroyd *)

PARI

\\ here b(n) is A056493
b(n) = sumdiv(n, d, moebius(n/d) * if(d%2,2,3) * 2^((d-1)\2));
a(n) = sumdiv(n, d, d*b(d)) / 2; \\ Andrew Howroyd, Oct 07 2017

Formula

a(n) = Sum_{d|n} d*A056493(d)/2. - Andrew Howroyd, Oct 07 2017

From Petros Hadjicostas, Oct 15 2017: (Start)

a(n) = (n/2)*Sum_{d|n} (phi^(-1)(d)/d)*b(n/d), where phi^(-1)(n) = A023900(n) is the Dirichlet inverse of the Euler totient function and b(n) = A029744(n+1) (= 3*2^((n/2)-1), if n is even, and = 2^((n+1)/2), if n is odd).

G.f.: Sum_{n>=1} phi^(-1)(n)*g(x^n), where phi^(-1)(n) = A023900(n) and g(x) = x*(x+1)*(2*x+1)/(1-2*x^2)^2.

(End)

Extensions

a(11) - a(24) from Robert G. Wilson v, Aug 25 2010

a(25) - a(27) from Robert G. Wilson v, Aug 29 2010

Terms a(28) and beyond from Andrew Howroyd, Oct 07 2017