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

Author

David W. Wilson

Keywords

• nonn

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

• Joerg Arndt, Matters Computational (The Fxtbook), p.119
• Index entries for sequences related to Lyndon words

Crossrefs

Cf. A000740, A045669, A045678, A045683.

Programs

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 *)

Formula

From Andrew Howroyd, Sep 28 2017: (Start)

Moebius transform of A045678.

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

(End)

Extensions

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