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

0,4

Comments

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

Links

Table of n, a(n) for n=0..27.

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.

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.

Sequence in context: A317020 A141499 A264553 * A219709 A061772 A032165

Adjacent sequences: A278638 A278639 A278640 * A278642 A278643 A278644

Keyword

nonn

Author

Herbert Kociemba, Nov 24 2016

Status

approved