A045687 -id:A045687 - cp's OEIS frontend

A045669 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement, inequivalent to reverse and complement.

0, 0, 0, 12, 32, 120, 288, 784, 1792, 4284, 9600, 21824, 47520, 104832, 225792, 487260, 1040384, 2219520, 4699296, 9942016, 20930400, 43994748, 92184576, 192843776, 402451200, 838655400, 1744404480, 3623423328, 7515275040

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A045664, A045665, A045687.

a(n) = 2*n*A045687(n).

a(n) = A045664(n) - A045665(n). - Andrew Howroyd, Sep 14 2019

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.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

nonn

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

Index entries for sequences related to Lyndon words

Cf. A000048, A045677, A045683, A045687.

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[#]&];
a[n_] := If[n == 0, 0, a48[n] - a45683[n]];
a /@ Range[0, 36] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)

From Andrew Howroyd, Sep 28 2017: (Start)
Moebius transform of A045677.
a(n) = A000048(n) - A045683(n).
(End)

cp's OEIS Frontend

A045669 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement, inequivalent to reverse and complement.

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