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

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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

Crossrefs

Cf. A000013, A003239, A011782, A045674, A128014.

Programs

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];
a /@ Range[0, 100] (* Jean-François Alcover, Sep 23 2019 *)

Formula

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