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%I #14 Sep 23 2019 13:25:11
%S 0,0,0,0,0,8,32,168,616,2380,8472,30760,109644,394816,1420784,5149948,
%T 18736744,68553728,251902032,929814984,3445433608,12814382620,
%U 47817551136,178982546512,671813695340,2528191984504,9536849826816
%N Number of 2n-bead balanced binary necklaces which are not equivalent to their reverse, complement or reversed complement.
%C The number of 2n-bead balanced binary necklaces is A003239(n). The number which are equivalent to their reverse, complement and reversed complement are respectively A128014(n), A000013(n) and A011782(n). - _Andrew Howroyd_, Sep 28 2017
%H Jean-François Alcover, <a href="/A045675/b045675.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/Ne#necklaces">Index entries for sequences related to necklaces</a>
%F a(n) = A003239(n) - A128014(n) - A000013(n) - A011782(n) + 2*A045674(n). - _Andrew Howroyd_, Sep 28 2017
%t a3239[n_] := If[n==0, 1, Sum[EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]];
%t a128014[n_] := SeriesCoefficient[(1 + x)/Sqrt[1 - 4 x^2], {x, 0, n}];
%t a11782[n_] := SeriesCoefficient[(1 - x)/(1 - 2x), {x, 0, n}];
%t a13[n_] := If[n==0, 1, Sum[(EulerPhi[2d]*2^(n/d)), {d, Divisors[n]}]/(2n)];
%t a45674[n_] := a45674[n] = If[n==0, 1, If[EvenQ[n], 2^(n/2-1) + a45674[n/2], 2^((n-1)/2)]];
%t a[n_] := a3239[n] - a128014[n] - a13[n] - a11782[n] + 2 a45674[n];
%t a /@ Range[0, 100] (* _Jean-François Alcover_, Sep 23 2019 *)
%Y Cf. A000013, A003239, A011782, A045674, A128014.
%K nonn
%O 0,6
%A _David W. Wilson_
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