login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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.
1
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
0,7
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
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[#]&];
a[n_] := If[n == 0, 0, a48[n] - a45683[n]];
a /@ Range[0, 36] (* Jean-François Alcover, Sep 23 2019, after Andrew Howroyd *)
CROSSREFS
KEYWORD
nonn
STATUS
approved