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A032241 Number of identity bracelets of n beads of 4 colors. 1
4, 6, 4, 15, 72, 266, 1044, 3780, 14056, 51132, 188604, 693845, 2572920, 9566046, 35758628, 134134080, 505159200, 1908539864, 7233104844, 27486455049, 104713295712, 399817073946, 1529746919604 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..23.

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

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)

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 *)

CROSSREFS

Sequence in context: A019189 A019190 A143174 * A065748 A019077 A019245

Adjacent sequences:  A032238 A032239 A032240 * A032242 A032243 A032244

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified June 16 12:38 EDT 2019. Contains 324152 sequences. (Running on oeis4.)