A032280 Number of bracelets (turnover necklaces) of n beads of 2 colors, 7 of them black.

1, 1, 4, 8, 20, 38, 76, 133, 232, 375, 600, 912, 1368, 1980, 2829, 3936, 5412, 7293, 9724, 12760, 16588, 21287, 27092, 34112, 42640, 52819, 65008, 79392, 96405, 116280, 139536, 166464, 197676, 233529, 274740, 321741, 375364

Offset: 7

Author

Christian G. Bower

Keywords

• nonn
• easy

Comments

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of nonequivalent necklaces of 7 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=7 (see our comment to A032279).

(End)

References

• N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 7..1000
• C. G. Bower, Transforms (2)
• S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
• Hansraj 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.
• F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
• Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
• Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
• Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
• Vladimir 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
• Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,1,0,-1,8,-6,-6,8,0,-3,1).

Crossrefs

Column k=7 of A052307.

Programs

Mathematica

k = 7; 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[-(4 x^6 - 2 x^5 - 2 x^4 + 4 x^3 + x^2 - 2 x + 1)/((x - 1)^7 (x + 1)^3 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 19 2013 *)
k=7; 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 *)

Formula

S. J. Cyvin et al. (1997) give a g.f.

"DIK[ 7 ]" (necklace, indistinct, unlabeled, 7 parts) transform of 1, 1, 1, 1...

From Vladimir Shevelev, Apr 23 2011: (Start)

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

a(n) = (3/7)*s(n,0,7) + (48*C(n-1,6) + 7*(n-2)*(n-4)*(n-6))/672, if n is even;

a(n) = (3/7)*s(n,0,7) + (48*C(n-1,6) + 7*(n-1)*(n-3)*(n-5))/672, if n is odd. (End)

G.f.: -x^7*(4*x^6-2*x^5-2*x^4+4*x^3+x^2-2*x+1) / ((x-1)^7*(x+1)^3*(x^6+x^5+x^4+x^3+x^2+x+1)). - Colin Barker, Feb 06 2013

From Herbert Kociemba, Nov 05 2016: (Start)

G.f.: (1/2)*x^7*((1+x)/(1-x^2)^4 + 1/7*(1/(1-x)^7 + 6/(1-x^7))).

G.f.: k=7, 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)