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A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.

%I A006840 M0837 #50 Dec 01 2017 18:58:16
%S A006840 1,1,2,3,7,13,35,85,257,765,2518,8359,28968,101340,361270,1297879,
%T A006840 4707969,17179435,63068876,232615771,861725794,3204236779,11955836258,
%U A006840 44748176653,167959144032,632058070310,2384235077576,9013628451275
%N A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.
%D A006840 J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89.
%D A006840 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006840 G. C. Greubel, <a href="/A006840/b006840.txt">Table of n, a(n) for n = 0..1650</a> (terms 0..200 from Andrew Howroyd)
%H A006840 J. A. Hoskins, C. E. Praeger and A. P. Street, <a href="/A006840/a006840.pdf">Balanced twills with bounded float length</a>, Congress. Numerantium, 40 (1983), 77-89. (Annotated scanned copy)
%H A006840 W. D. Hoskins and Anne Penfold Street, <a href="http://dx.doi.org/10.1017/S1446788700017547">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
%H A006840 W. D. Hoskins and A. P. Street, <a href="/A005513/a005513_1.pdf">Twills on a given number of harnesses</a>, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
%H A006840 A. P. Street, <a href="/A005513/a005513.pdf">Letter to N. J. A. Sloane, N.D.</a>
%H A006840 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%F A006840 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_
%t A006840 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_ *)
%o A006840 (PARI) \\ here b is A045629
%o A006840 b(n) = (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1));
%o A006840 a(n) = if(n==0, 1, (b(n) + 2^(n-2) + binomial(n-n%2, n\2)/2) / 2); \\ _Andrew Howroyd_, Sep 27 2017
%Y A006840 Row sums of A259341.
%Y A006840 Cf. A045629.
%K A006840 nonn
%O A006840 0,3
%A A006840 _N. J. A. Sloane_
%E A006840 More terms from _David W. Wilson_