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%I #25 Sep 28 2018 10:02:13
%S 6,21,56,231,888,4291,20646,107331,563786,3037314,16514106,90782986,
%T 502474356,2799220041,15673673176,88162676511,497847963696,
%U 2821127825971,16035812864946,91404068329560
%N Number of bracelets of length n using a maximum of six different colored beads.
%C Turning over will not create a new bracelet.
%D M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]
%F a(n) = Sum_{d|n} phi(d)*6^(n/d)/(2*n);
%F a(n) = 6^((n+1)/2)/2 for n odd,
%F (7/4)*6^(n/2) for n even.
%F G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 6*x^n)/n + (1+6*x+15*x^2)/(1-6*x^2))/2. - _Herbert Kociemba_, Nov 02 2016
%e For n=2, the 21 bracelets are AA, AB, AC, AD, AE, AF, BB, BC, BD, BE, BF, CC, CD, CE, CF, DD, DE, DF, EE, EF, and FF. - _Robert A. Russell_, Sep 24 2018
%t mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-6*x^n]/n,{n,mx}]+(1+6 x+15 x^2)/(1-6 x^2))/2,{x,0,mx}],x] (* _Herbert Kociemba_, Nov 02 2016 *)
%t k=6; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* _Robert A. Russell_, Sep 24 2018 *)
%Y Cf. A000029, A054625.
%Y Cf. a(n) = A081720(n,6), n >= 6. - _Wolfdieter Lang_, Jun 03 2012
%Y Column 6 of A051137.
%Y Equals (A054625 + A056488) / 2 = A054625 - A278642 = A278642 + A056488.
%K nonn
%O 1,1
%A _Marks R. Nester_
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