A045687 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reversed complement, but are not equivalent to their reverse and complement.

0, 0, 0, 2, 4, 12, 24, 56, 112, 238, 480, 992, 1980, 4032, 8064, 16242, 32512, 65280, 130536, 261632, 523260, 1047494, 2095104, 4192256, 8384400, 16773108, 33546240, 67100432, 134201340, 268419072, 536837640, 1073709056, 2147418112

Offset

0,4

Comments

The number of length 2n balanced binary Lyndon words which are equivalent to their reversed complement is A000740(n) and the number which are equivalent to their reverse, complement and reversed complement is A045683(n). - Andrew Howroyd, Sep 28 2017

Links

Table of n, a(n) for n=0..32.

Joerg Arndt, Matters Computational (The Fxtbook), p.119

Index entries for sequences related to Lyndon words

Formula

From Andrew Howroyd, Sep 28 2017: (Start)

Moebius transform of A045678.

a(n) = A000740(n) - A045683(n).

(End)

Mathematica

a740[n_] := DivisorSum[n, MoebiusMu[n/#]*2^(#-1)&];
a45674[0] = 1; a45674[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r-1)]; t + 2^Quotient[r, 2]];
a45683[0] = 1; a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*a45674[#]&];
a[0] = 0; a[n_] := a740[n] - a45683[n];
Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Sep 30 2017, after Andrew Howroyd *)

Crossrefs

Cf. A000740, A045669, A045678, A045683.

Sequence in context: A133411 A201078 A004645 * A233411 A057422 A036045

Adjacent sequences: A045684 A045685 A045686 * A045688 A045689 A045690

Keyword

nonn

Author

David W. Wilson

Extensions

Incorrect formulas and comments removed by Andrew Howroyd, Sep 28 2017

Status

approved