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

0,6

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

Crossrefs

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

Sequence in context: A200153 A188121 A045684 * A086348 A129798 A129792

Adjacent sequences: A045672 A045673 A045674 * A045676 A045677 A045678

Keyword

nonn

Author

David W. Wilson

Status

approved