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A032281 Number of bracelets (turnover necklaces) of n beads of 2 colors, 9 of them black. 5
1, 1, 5, 12, 35, 79, 185, 375, 750, 1387, 2494, 4262, 7105, 11410, 17930, 27407, 41107, 60335, 87154, 123695, 173173, 238957, 325845, 438945, 585265, 772252, 1009868, 1308742, 1682660, 2146420, 2718806, 3419924, 4274905 (list; graph; refs; listen; history; text; internal format)
OFFSET

9,3

COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 9 beads each of them painted by one of n colors.

The sequence solves the so-called Reis problem about convex k-gons in case k=9 (see our comment to A032279).

(End)

LINKS

Table of n, a(n) for n=9..41.

C. G. Bower, Transforms (2)

H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

V. Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).

Index entries for sequences related to bracelets

FORMULA

"DIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...

Put s(n,k,d)=1, if n==k (mod d), and s(n,k,d)=0, otherwise. Then a(n)=(1/3)*s(n,0,9)+(n-3)*(n-6)*s(n,0,3)/162+(n-2)(n-4)*(n-6)*(n-8)*(945+(n-1)*(n-3)*(n-5)*(n-7))/725760, if n is even; a(n)=(1/3)*s(n,0,9)+(n-3)*(n-6)*s(n,0,3)/162+(n-1)*(n-3)*(n-5)*(n-7)*(945+(n-2)*(n-4)*(n-6)*(n-8))/725760, if n is odd. - Vladimir Shevelev, Apr 23 2011

From Herbert Kociemba, Nov 05 2016: (Start)

G.f.: 1/2*x^9*((1+x)/(1-x^2)^5 + 1/9*(1/(1-x)^9 - 2/(-1+x^3)^3 - 6/(-1+x^9))).

G.f.: k=9, x^k*((1/k)*Sum_{d|k} phi(d)*(1-x^d)^(-k/d) + (1+x)/(1-x^2)^floor[(k+2)/2])/2. [edited by Petros Hadjicostas, Jul 18 2018]

(End)

MATHEMATICA

k = 9; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)

k=9; CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2, {x, 0, 50}], x] (* Herbert Kociemba, Nov 04 2016 *)

CROSSREFS

Sequence in context: A298992 A050189 A116995 * A294654 A229043 A185699

Adjacent sequences:  A032278 A032279 A032280 * A032282 A032283 A032284

KEYWORD

nonn

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)