A278640 Number of pairs of orientable necklaces with n beads and up to 4 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, 4, 15, 72, 270, 1044, 3795, 14060, 51204, 188604, 694130, 2572920, 9567090, 35758704, 134137875, 505159200, 1908554190, 7233104844, 27486506268, 104713296760, 399817262550, 1529746919604, 5864041395730, 22517964582504, 86607602546220, 333599838189804, 1286742419927070, 4969488707124120, 19215357085867800

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..30.

Formula

G.f.: k=4, (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=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

Example

Example: For 3 beads and the colors A, B, C and D the 4 orientable necklaces are ABC, ABD, ACD and BCD. The turned-over necklaces ACB, ADB, ADC and BDC are not included in the count.

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, 4], {x, 0, mx}], x]
k=4; 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 4 of A293496.

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

Equals (A001868 - A056486) / 2 = A001868 - A032275 = A032275 - A056486.

Sequence in context: A027316 A085349 A185133 * A026992 A039764 A171005

Adjacent sequences: A278637 A278638 A278639 * A278641 A278642 A278643

Keyword

nonn

Author

Herbert Kociemba, Nov 24 2016

Status

approved