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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.
3

%I #25 Sep 30 2017 16:45:54

%S 0,0,0,2,4,12,24,56,112,238,480,992,1980,4032,8064,16242,32512,65280,

%T 130536,261632,523260,1047494,2095104,4192256,8384400,16773108,

%U 33546240,67100432,134201340,268419072,536837640,1073709056,2147418112

%N 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 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 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, p.119

%H <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>

%F From _Andrew Howroyd_, Sep 28 2017: (Start)

%F Moebius transform of A045678.

%F a(n) = A000740(n) - A045683(n).

%F (End)

%t a740[n_] := DivisorSum[n, MoebiusMu[n/#]*2^(#-1)&];

%t 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 a45683[0] = 1; a45683[n_] := DivisorSum[n, MoebiusMu[n/#]*a45674[#]&];

%t a[0] = 0; a[n_] := a740[n] - a45683[n];

%t Table[a[n], {n, 0, 32}] (* _Jean-François Alcover_, Sep 30 2017, after _Andrew Howroyd_ *)

%Y Cf. A000740, A045669, A045678, A045683.

%K nonn

%O 0,4

%A _David W. Wilson_

%E Incorrect formulas and comments removed by _Andrew Howroyd_, Sep 28 2017