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A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.
(Formerly M0837)
9
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A056953 A321681 A045611 * A123408 A033995 A013917

Adjacent sequences:  A006837 A006838 A006839 * A006841 A006842 A006843

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

STATUS

approved

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Last modified December 14 14:20 EST 2019. Contains 329979 sequences. (Running on oeis4.)