A045629 Number of 2n-bead black-white complementable necklaces with n black beads.

1, 1, 2, 3, 7, 15, 44, 128, 415, 1367, 4654, 16080, 56450, 200170, 716728, 2585850, 9393119, 34319667, 126047906, 465076160, 1723097066, 6407856892, 23910271224, 89493903438, 335912741682, 1264106399934, 4768448177636, 18027218147818

Offset

0,3

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for sequences related to necklaces

Formula

a(n) = (1/2n) * Sum_{d|n} (phi(n/d)*C(2d-1, d-1) + phi(2n/d)*2^(d-1)). - Christian G. Bower

a(n) ~ 4^(n-1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 08 2017

Mathematica

a[n_] := If[n == 0, 1, (1/(2*n))*DivisorSum[n, EulerPhi[n/#1]*Binomial[2*#1 - 1, #1 - 1] + EulerPhi[2*(n/#1)]*2^(#1 - 1)&]];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

Prog

(PARI) a(n) = if(n==0, 1, (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1))); \\ Andrew Howroyd, Sep 27 2017

Crossrefs

Cf. A006840.

Sequence in context: A368564 A213385 A161746 * A034731 A216435 A110888

Adjacent sequences: A045626 A045627 A045628 * A045630 A045631 A045632

Keyword

nonn

Author

David W. Wilson

Status

approved