OFFSET
1,1
COMMENTS
Turning over will not create a new bracelet.
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.]
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
EXAMPLE
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
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 *)
CROSSREFS
Cf. a(n) = A081720(n,6), n >= 6. - Wolfdieter Lang, Jun 03 2012
Column 6 of A051137.
KEYWORD
nonn
AUTHOR
STATUS
approved