A278641 Number of pairs of orientable necklaces with n beads and up to 5 colors; i.e., turning the necklace over does not leave it unchanged. The turned-over necklace is not included in the count.

0, 0, 0, 10, 45, 252, 1130, 5270, 23520, 106960, 483756, 2211650, 10149805, 46911060, 217868310, 1017057518, 4767797895, 22438419120, 105960938380, 501928967930, 2384171386941, 11353241261180, 54185968572450, 259150507387910, 1241763071712930, 5960463867187752, 28656077411358180, 137973711706163210

Offset: 0

Author

Herbert Kociemba, Nov 24 2016

Keywords

• nonn

Comments

Number of chiral bracelets of n beads using up to five different colors.

Crossrefs

Column 5 of A293496.

Cf. A059076 (2 colors), A278639 (3 colors), A278640 (4 colors).

a(n) = (A001869(n) - A056487(n+1)) / 2 = A032276(n) - A056487(n+1).

Equals A001869 - A032276.

Programs

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 *)

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