

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
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OFFSET

5,3


COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of nonequivalent necklaces of 5 beads each of them painted by one of n colors.
The sequence solves the socalled Reis problem about convex kgons 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 symmetryallowed, linearlyindependent terms at nth 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 nonordered nonisometric convex quadrilaterals inscribed in a regular ngon, 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), 867873.
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 kgons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964999.
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 kgons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629638.
V. Shevelev, A problem of enumeration of twocolor bracelets with several variations, arXiv:0710.1370 [math.CO], 20072011.
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*(1x+2*x^3x^5+x^6)/((1x)^2*(1x^2)^2*(1x^5)).
Put s(n,k,d)=1, if n==k(mod d), 0, otherwise. Then
a(n) = 0.4*s(n,0,5)+(n1)*(n3)*((n2)*(n4)+15)/240, if n is odd; a(n)=0.4*s(n,0,5)+(n2)*(n4)*((n1)*(n3)+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[#] (1x^#)^((k/#))&/@Divisors[k])+(1+x)/(1x^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*(1n%2))*n)/240 +3/5) \\ Washington Bomfim, Jul 17 2008
(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(1x+2*x^3x^5+x^6)/((1x)^2*(1x^2)^2*(1x^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



