OFFSET
0,4
COMMENTS
Number of chiral bracelets of n beads using up to five different colors.
FORMULA
G.f.: k=5, (1 - Sum_{n>=1} phi(n)*log(1 - k*x^n)/n - Sum_{i=0..2} Binomial[k,i]*x^i / ( 1-k*x^2) )/2.
For n>0, a(n) = -(k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/2n)* Sum_{d|n} phi(d)*k^(n/d), where k=5 is the maximum number of colors. - Robert A. Russell, Sep 24 2018
MATHEMATICA
mx=40; f[x_, k_]:=(1-Sum[EulerPhi[n]*Log[1-k*x^n]/n, {n, 1, mx}]-Sum[Binomial[k, i]*x^i, {i, 0, 2}]/(1-k*x^2))/2; CoefficientList[Series[f[x, 5], {x, 0, mx}], x]
k=5; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) - (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 0] (* Robert A. Russell, Sep 24 2018 *)
CROSSREFS
Column 5 of A293496.
KEYWORD
nonn
AUTHOR
Herbert Kociemba, Nov 24 2016
STATUS
approved