A032241 Number of identity bracelets of n beads of 4 colors.

4, 6, 4, 15, 72, 266, 1044, 3780, 14056, 51132, 188604, 693845, 2572920, 9566046, 35758628, 134134080, 505159200, 1908539864, 7233104844, 27486455049, 104713295712, 399817073946, 1529746919604

Offset: 1

Author

Christian G. Bower

Keywords

• nonn

Comments

For n>2 also number of asymmetric bracelets with n beads of four colors. - Herbert Kociemba, Nov 29 2016

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• C. G. Bower, Transforms (2)
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Index entries for sequences related to bracelets

Crossrefs

Column k=4 of A309528 for n >= 3.

Programs

Mathematica

m = 4; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d] (m^(n/d)/n - If[OddQ[n/d], m^((n/d + 1)/2), ((m + 1) m^(n/(2 d))/2)]), {d, Divisors[n]}]/2, {n, 3, 20}] (* Robert A. Russell, Mar 18 2013 *)
mx=40;gf[x_,k_]:=Sum[MoebiusMu[n]*(-Log[1-k*x^n]/n-Sum[Binomial[k,i]x^(n i),{i,0,2}]/(1-k x^(2n)))/2,{n,mx}];ReplacePart[Rest[CoefficientList[Series[gf[x,4],{x,0,mx}],x]],{1->4,2->6}] (* Herbert Kociemba, Nov 29 2016 *)

PARI

a(n)={if(n<3, binomial(4, n), sumdiv(n, d, moebius(n/d)*(4^d/n - if(d%2, 4^((d+1)/2), 5*4^(d/2)/2)))/2)} \\ Andrew Howroyd, Sep 12 2019

Formula

"DHK" (bracelet, identity, unlabeled) transform of 4, 0, 0, 0...

From Herbert Kociemba, Nov 29 2016: (Start)

More generally, gf(k) is the g.f. for the number of asymmetric bracelets with n beads of k colors.

gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n - Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)