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A005515 Number of n-bead bracelets (turnover necklaces) of two colors with 10 red beads and n-10 black beads.
(Formerly M4105)
3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

10,3

COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent (turnover) necklaces of 10 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=10 (see our comment to A032279).

(End)

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

Table of n, a(n) for n=10..40.

H. 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)

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]

V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.

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).

A. P. Street, Letter to N. J. A. Sloane, N.D.

Index entries for sequences related to bracelets

FORMULA

From Vladimir Shevelev, Apr 23 2011: (Start)

Put s(n,k,d)=1, if n==k(mod d), 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} EulerPhi(d)*(1-x^d)^(-(k/d)) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2. [edited by Petros Hadjicostas, Jan 10 2019]

(End)

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 *)

CROSSREFS

Sequence in context: A093369 A130443 A294655 * A114705 A200187 A107301

Adjacent sequences:  A005512 A005513 A005514 * A005516 A005517 A005518

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

Sequence extended and description corrected by Christian G. Bower

Name edited by Petros Hadjicostas, Jan 10 2019

STATUS

approved

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Last modified January 19 09:35 EST 2019. Contains 319306 sequences. (Running on oeis4.)