A045675 - OEIS

A045675

Number of 2n-bead balanced binary necklaces which are not equivalent to their reverse, complement or reversed complement.

0, 0, 0, 0, 0, 8, 32, 168, 616, 2380, 8472, 30760, 109644, 394816, 1420784, 5149948, 18736744, 68553728, 251902032, 929814984, 3445433608, 12814382620, 47817551136, 178982546512, 671813695340, 2528191984504, 9536849826816

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OFFSET

0,6

COMMENTS

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

LINKS

Jean-François Alcover, Table of n, a(n) for n = 0..100

Index entries for sequences related to necklaces

FORMULA

a(n) = A003239(n) - A128014(n) - A000013(n) - A011782(n) + 2*A045674(n). - Andrew Howroyd, Sep 28 2017

MATHEMATICA

a3239[n_] := If[n==0, 1, Sum[EulerPhi[n/k]*Binomial[2k, k]/(2n), {k, Divisors[n]}]];

a128014[n_] := SeriesCoefficient[(1 + x)/Sqrt[1 - 4 x^2], {x, 0, n}];

a11782[n_] := SeriesCoefficient[(1 - x)/(1 - 2x), {x, 0, n}];

a13[n_] := If[n==0, 1, Sum[(EulerPhi[2d]*2^(n/d)), {d, Divisors[n]}]/(2n)];

a45674[n_] := a45674[n] = If[n==0, 1, If[EvenQ[n], 2^(n/2-1) + a45674[n/2], 2^((n-1)/2)]];

a[n_] := a3239[n] - a128014[n] - a13[n] - a11782[n] + 2 a45674[n];