A005513 Number of n-bead bracelets (turnover necklaces) of two colors with 6 red beads and n-6 black beads.
1, 1, 4, 7, 16, 26, 50, 76, 126, 185, 280, 392, 561, 756, 1032, 1353, 1782, 2277, 2920, 3652, 4576, 5626, 6916, 8372, 10133, 12103, 14448, 17063, 20128, 23528, 27474, 31824, 36822, 42315, 48564, 55404, 63133, 71554, 81004
Offset: 6
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- 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 = 6..5000
- 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.
- W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
- W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
- A. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188
- 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, 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)
- A. P. Street, Letter to N. J. A. Sloane, N.D.
- Index entries for sequences related to bracelets
- Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).
Crossrefs
Column k=6 of A052307.
Programs
-
Maple
A005513 := proc(n) if n mod 6 = 0 then (24*binomial(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288 elif n mod 6 = 3 then (24*binomial(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288 elif n mod 6 = 2 or n mod 6 = 4 then (8*binomial(n-1,5)+(n+1)*(n-2)*(n-4))/96 elif n mod 6 = 1 or n mod 6 = 5 then (8*binomial(n-1,5)+(n-1)*(n-3)*(n-5))/96 fi: end: seq(A005513(n),n=6..44); # Johannes W. Meijer, Aug 11 2011
-
Mathematica
k = 6; 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=6; 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 *) CoefficientList[Series[(1/12) (1/(1 - x)^6 + 4/(1 - x^2)^3 + 2/(1 - x^3)^2 + 3/((1 - x)^2 (1 - x^2)^2) + 2/(1 - x^6)), {x, 0, 43}], x] (* Vincenzo Librandi, Apr 24 2018 *)
Formula
S. J. Cyvin et al. (1997) give a g.f.
G.f.: (x^6/12)*(1/(1-x)^6+4/(1-x^2)^3+2/(1-x^3)^2+3/((1-x)^2*(1-x^2)^2)+2/(1-x^6)). - Vladeta Jovovic, Feb 28 2007
G.f.: x^6*(1-x+x^2+x^3+2*x^4+2*x^6+x^8-x^5) / ( (x^2-x+1)*(1+x+x^2)^2*(1+x)^3*(x-1)^6 ). - R. J. Mathar, Sep 18 2011
From Vladimir Shevelev, Apr 23 2011: (Start)
if n==0 mod 6, a(n)=(24*C(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288;
if n==3 mod 6, a(n)=(24*C(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288;
if n==2,4 mod 6, a(n)=(8*C(n-1,5)+(n+1)*(n-2)*(n-4))/96;
if n==1,5 mod 6, a(n)=(8*C(n-1,5)+(n-1)*(n-3)*(n-5))/96.
(End)
Extensions
Sequence extended and description corrected by Christian G. Bower
Name edited by Petros Hadjicostas, Jan 10 2019
Comments