A045686 - OEIS

A045686

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

0, 0, 0, 0, 0, 0, 2, 2, 8, 14, 36, 62, 140, 252, 522, 968, 1920, 3600, 7028, 13286, 25704, 48914, 94302, 180314, 347480, 666996, 1286460, 2477328, 4785300, 9240012, 17879314, 34604066, 67076096, 130084990, 252579600, 490722342, 954306080

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OFFSET

0,7

COMMENTS

The number of length 2n balanced binary Lyndon words which are equivalent to their complement is A000048(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..36.

Index entries for sequences related to Lyndon words

FORMULA

From Andrew Howroyd, Sep 28 2017: (Start)

Moebius transform of A045677.

a(n) = A000048(n) - A045683(n).

(End)

MATHEMATICA

a48[n_] := Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n);

b[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]; t + 2^Quotient[r, 2]];

a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*b[#]&];