

A006840


Number of 2nbead blackwhite 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
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OFFSET

0,3


REFERENCES

J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 7789.
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), 7789. (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, 115.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 115. (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(n1, (n1)/2) + 2^(n2)); if n is even, a(n) = (1/2) * (A045629 + (1/2) * C(n, n/2) + 2^(n2)).  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^(n2) + Binomial[n  Mod[n, 2], Quotient[n, 2]]/2)/2; Table[a[n], {n, 0, 27}] (* JeanFrançois 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*d1, d1) + eulerphi(2*n/d)*2^(d1));
a(n) = if(n==0, 1, (b(n) + 2^(n2) + binomial(nn%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



