A045684 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are inequivalent to their reverse, complement and reversed complement.

0, 0, 0, 0, 0, 8, 32, 168, 616, 2380, 8464, 30760, 109612, 394816, 1420616, 5149940, 18736128, 68553728, 251899620, 929814984, 3445425136, 12814382452, 47817520376, 178982546512, 671813585080, 2528191984496, 9536849432000

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

The number of length 2n balanced binary Lyndon words is A022553(n) and the number which are equivalent to their reverse, complement and reversed complement are respectively A045680(n), A000048(n) and A000740(n). - Andrew Howroyd, Sep 29 2017

Links

• Jean-François Alcover, Table of n, a(n) for n = 0..100
• Index entries for sequences related to Lyndon words

Crossrefs

Cf. A000048, A000740, A022553, A045675, A045680, A045683.

Programs

Mathematica

a22553[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d]*Binomial[2d, d], {d, Divisors[n]}]/(2n)];
a45680[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#] Binomial[# - Mod[#, 2], Quotient[#, 2]] &]];
a48[n_] := If[n == 0, 1, Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n)];
a740[n_] := Sum[MoebiusMu[n/d]*2^(d - 1), {d, Divisors[n]}];
b[n_] := Module[{t = 0, r = n}, If[n == 0, 1, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]]; t + 2^Quotient[r, 2]];
a45683[n_] := If[n == 0, 1, DivisorSum[n, MoebiusMu[n/#]*b[#] &]];
a[n_] := If[n == 0, 0, a22553[n] - a45680[n] - a48[n] - a740[n] + 2 a45683[n]];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 23 2019 *)

Formula

From Andrew Howroyd, Sep 28 2017: (Start)

Moebius transform of A045675.

a(n) = A022553(n) - A045680(n) - A000048(n) - A000740(n) + 2*A045683(n).

(End)