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A045687 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reversed complement, but are not equivalent to their reverse and complement.

%I A045687 #25 Sep 30 2017 16:45:54
%S A045687 0,0,0,2,4,12,24,56,112,238,480,992,1980,4032,8064,16242,32512,65280,
%T A045687 130536,261632,523260,1047494,2095104,4192256,8384400,16773108,
%U A045687 33546240,67100432,134201340,268419072,536837640,1073709056,2147418112
%N A045687 Number of 2n-bead balanced binary necklaces of fundamental period 2n which are equivalent to their reversed complement, but are not equivalent to their reverse and complement.
%C A045687 The number of length 2n balanced binary Lyndon words which are equivalent to their reversed complement is A000740(n) and the number which are equivalent to their reverse, complement and reversed complement is A045683(n). - _Andrew Howroyd_, Sep 28 2017
%H A045687 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p.119
%H A045687 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A045687 From _Andrew Howroyd_, Sep 28 2017: (Start)
%F A045687 Moebius transform of A045678.
%F A045687 a(n) = A000740(n) - A045683(n).
%F A045687 (End)
%t A045687 a740[n_] := DivisorSum[n, MoebiusMu[n/#]*2^(#-1)&];
%t A045687 a45674[0] = 1; a45674[n_] := Module[{t = 0, r = n}, While[EvenQ[r], r = Quotient[r, 2]; t += 2^(r-1)]; t + 2^Quotient[r, 2]];
%t A045687 a45683[0] = 1; a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*a45674[#]&];
%t A045687 a[0] = 0; a[n_] := a740[n] - a45683[n];
%t A045687 Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Sep 30 2017, after _Andrew Howroyd_ *)
%Y A045687 Cf. A000740, A045669, A045678, A045683.
%K A045687 nonn
%O A045687 0,4
%A A045687 _David W. Wilson_
%E A045687 Incorrect formulas and comments removed by _Andrew Howroyd_, Sep 28 2017