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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.

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 February 24 14:41 EST 2018. Contains 299623 sequences. (Running on oeis4.)