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A032279 Number of bracelets (turn over necklaces) of n beads of 2 colors, 5 of them black. 14
1, 1, 3, 5, 10, 16, 26, 38, 57, 79, 111, 147, 196, 252, 324, 406, 507, 621, 759, 913, 1096, 1298, 1534, 1794, 2093, 2421, 2793, 3199, 3656, 4152, 4706, 5304, 5967, 6681, 7467, 8311, 9234, 10222, 11298, 12446, 13691, 15015, 16445 (list; graph; refs; listen; history; text; internal format)
OFFSET

5,3

COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 5 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=5. The full solution was given by H.Gupta (1979); I gave a short proof of Gupta's result and showed an equivalence of this problem and every of the following problems: enumerating the bracelets of n beads of 2 colors, k of them black, and enumerating the necklaces of k beads each of them painted by one of n colors.

a(n) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order n with five 1's in every row.

(End)

a(n+5) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T_1 X h vibronic perturbation matrix, H(Q) (cf. Dunn & Bates). - Bradley Klee, Jul 20 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 5..1000

N. Benyahia Tani, Z. Yahi, S. Bouroubi, Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon, Bulletin du Laboratoire Liforce, 01 (2014) 1 - 9.

C. G. Bower, Transforms (2)

S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of PĆ³lya's theorem, Z. Naturforsch., 52a (1997), 867-873.

J. L. Dunn and C. A. Bates, Analysis of the T1u(x)hg system as a model for C60 molecules, Phys. Rev. B 52, 5996, 15 August 1995.

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

E. Kirkman, J. Kuzmanovich and J. J. Zhang, Invariants of (-1)-Skew Polynomial Rings under Permutation Representations, arXiv preprint arXiv:1305.3973, 2013. See Example 5.5.

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, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.

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[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of 1, 1, 1, 1, ...

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

Put s(n,k,d)=1, if n==k(mod d), 0, otherwise. Then

a(n) = 0.4*s(n,0,5)+(n-1)*(n-3)*((n-2)*(n-4)+15)/240, if n is odd; a(n)=0.4*s(n,0,5)+(n-2)*(n-4)*((n-1)*(n-3)+15)/240, if n is even. - Vladimir Shevelev, Apr 23 2011

a(n+5) = floor(n^4/240 + n^3/24 + 5*n^2/24 + 25*n/48 + 1 + (-1)^n*n/16). - Robert Israel, Jul 22 2015

MAPLE

seq(floor(n^4/240 + n^3/24 + 5*n^2/24 + 25*n/48 + 1 + (-1)^n*n/16), n=0..100); # Robert Israel, Jul 22 2015

MATHEMATICA

k = 5; 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 *)

CoefficientList[Series[(1 - x + 2 x^3 - x^5 + x^6) / ((1 - x)^2 (1 - x^2)^2 (1 - x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 07 2013 *)

k=5 (* Number of black beads in bracelet problem *); 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 *)

PROG

(PARI) a(n) = round((n^4 -10*n^3 +50*n^2 -(110+30*(1-n%2))*n)/240 +3/5) \\ Washington Bomfim, Jul 17 2008

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x+2*x^3-x^5+x^6)/((1-x)^2*(1-x^2)^2*(1-x^5))); // Vincenzo Librandi, Sep 07 2013

CROSSREFS

Cf. A008805.

Sequence in context: A184800 A267151 A209008 * A070558 A233758 A253769

Adjacent sequences:  A032276 A032277 A032278 * A032280 A032281 A032282

KEYWORD

nonn,easy,nice

AUTHOR

Christian G. Bower, N. J. A. Sloane

EXTENSIONS

G.f. corrected for offset 5 by Robert Israel, Jul 22 2015

STATUS

approved

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Last modified February 23 23:13 EST 2018. Contains 299595 sequences. (Running on oeis4.)