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

0,6

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

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)

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 *)

Crossrefs

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

Sequence in context: A336492 A200153 A188121 * A045675 A086348 A129798

Adjacent sequences: A045681 A045682 A045683 * A045685 A045686 A045687

Keyword

nonn

Author

David W. Wilson

Status

approved