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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.

%I A045686 #14 Sep 23 2019 09:45:07
%S A045686 0,0,0,0,0,0,2,2,8,14,36,62,140,252,522,968,1920,3600,7028,13286,
%T A045686 25704,48914,94302,180314,347480,666996,1286460,2477328,4785300,
%U A045686 9240012,17879314,34604066,67076096,130084990,252579600,490722342,954306080
%N 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.
%C A045686 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
%H A045686 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A045686 From _Andrew Howroyd_, Sep 28 2017: (Start)
%F A045686 Moebius transform of A045677.
%F A045686 a(n) = A000048(n) - A045683(n).
%F A045686 (End)
%t A045686 a48[n_] := Total[MoebiusMu[#]*2^(n/#)& /@ Select[Divisors[n], OddQ]]/(2n);
%t A045686 b[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r - 1)]; t + 2^Quotient[r, 2]];
%t A045686 a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*b[#]&];
%t A045686 a[n_] := If[n == 0, 0, a48[n] - a45683[n]];
%t A045686 a /@ Range[0, 36] (* _Jean-François Alcover_, Sep 23 2019, after _Andrew Howroyd_ *)
%Y A045686 Cf. A000048, A045677, A045683, A045687.
%K A045686 nonn
%O A045686 0,7
%A A045686 _David W. Wilson_