A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.
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
Keywords
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
Programs
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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 *) -
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
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
Extensions
More terms from David W. Wilson