A056341 Number of bracelets of length n using a maximum of six different colored beads.

6, 21, 56, 231, 888, 4291, 20646, 107331, 563786, 3037314, 16514106, 90782986, 502474356, 2799220041, 15673673176, 88162676511, 497847963696, 2821127825971, 16035812864946, 91404068329560

Offset: 1

Author

Marks R. Nester

Keywords

• nonn

Comments

Turning over will not create a new bracelet.

Examples

			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
		

References

• 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.]

Crossrefs

Cf. A000029, A054625.

Cf. a(n) = A081720(n,6), n >= 6. - Wolfdieter Lang, Jun 03 2012

Column 6 of A051137.

Equals (A054625 + A056488) / 2 = A054625 - A278642 = A278642 + A056488.

Programs

Mathematica

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 *)
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 *)

Formula

a(n) = Sum_{d|n} phi(d)*6^(n/d)/(2*n);

a(n) = 6^((n+1)/2)/2 for n odd,

(7/4)*6^(n/2) for n even.

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