A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads. (Formerly M0837)

1, 1, 2, 3, 7, 13, 35, 85, 257, 765, 2518, 8359, 28968, 101340, 361270, 1297879, 4707969, 17179435, 63068876, 232615771, 861725794, 3204236779, 11955836258, 44748176653, 167959144032, 632058070310, 2384235077576, 9013628451275

Offset

0,3

References

J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1650 (terms 0..200 from Andrew Howroyd)

J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89. (Annotated scanned copy)

W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.

W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)

A. P. Street, Letter to N. J. A. Sloane, N.D.

Index entries for sequences related to bracelets

Formula

If n is odd, a(n) = (1/2) * (A045629 + (1/2) * C(n-1, (n-1)/2) + 2^(n-2)); if n is even, a(n) = (1/2) * (A045629 + (1/2) * C(n, n/2) + 2^(n-2)). - Christian G. Bower

Mathematica

b[n_] := (1/(2*n))*DivisorSum[n, EulerPhi[n/#]*Binomial[2*# - 1, # - 1] + EulerPhi[2*(n/#)]*2^(# - 1) &]; a[0] = 1; a[n_] := (b[n] + 2^(n-2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

Prog

(PARI) \\ here b is A045629
b(n) = (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1));
a(n) = if(n==0, 1, (b(n) + 2^(n-2) + binomial(n-n%2, n\2)/2) / 2); \\ Andrew Howroyd, Sep 27 2017

Crossrefs

Row sums of A259341.

Cf. A045629.

Sequence in context: A371867 A321681 A045611 * A123408 A033995 A345247

Adjacent sequences: A006837 A006838 A006839 * A006841 A006842 A006843

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson

Status

approved