A005515 Number of n-bead bracelets (turnover necklaces) of two colors with 10 red beads and n-10 black beads.
1, 1, 6, 14, 47, 111, 280, 600, 1282, 2494, 4752, 8524, 14938, 25102, 41272, 65772, 102817, 156871, 235378, 346346, 502303, 716859, 1010256, 1404624, 1931540, 2625658, 3534776, 4711448, 6226148, 8156396, 10603704, 13679696, 17527595, 22304765, 28209566, 35459694
Offset: 10
Keywords
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
- Andrew Howroyd, Table of n, a(n) for n = 10..1000
- 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)
- Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
- Frank 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 (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).
Crossrefs
Column k=10 of A052307.
Programs
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Mathematica
k = 10; 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=10;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
From Vladimir Shevelev, Apr 23 2011: (Start)
Put s(n,k,d) = 1, if n == k (mod d), and s(n,k,d) = 0, otherwise. Then a(n) = n*s(n,0,5)/25 + (384*C(n-1,9) + (n+1)*(n-2)*(n-4)*(n-6)*(n-8))/7680, if n is even; a(n) = (n-5)*s(n,0,5)/25 + (384*C(n-1,9) + (n-1)*(n-3)*(n-5)*(n-7)*(n-9))/7680, if n is odd. (End)
From Herbert Kociemba, Nov 04 2016: (Start)
G.f.: (1/20)*x^10*(1/(-1+x)^10 + 10/((-1+x)^6*(1+x)^5) + 1/(1-x^2)^5 + 4/(-1+x^5)^2 - 4/(-1+x^10)).
G.f.: k=10, 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, Jan 10 2019] (End)
Extensions
Sequence extended and description corrected by Christian G. Bower
Name edited by Petros Hadjicostas, Jan 10 2019
Comments