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A032239 Number of identity bracelets of n beads of 2 colors. 2
2, 1, 0, 0, 0, 1, 2, 6, 14, 30, 62, 127, 252, 493, 968, 1860, 3600, 6902, 13286, 25446, 48914, 93775, 180314, 346420, 666996, 1284318, 2477328, 4781007, 9240012, 17870709, 34604066, 67058880, 130084990, 252545160, 490722342 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..35.

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 2, 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 = 2; (* 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, 2], {x, 0, mx}], x]], {1->2, 2->1}] (* Herbert Kociemba, Nov 29 2016 *)

CROSSREFS

Sequence in context: A110174 A022909 A292136 * A057094 A284938 A186084

Adjacent sequences:  A032236 A032237 A032238 * A032240 A032241 A032242

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 22 20:57 EDT 2018. Contains 316502 sequences. (Running on oeis4.)