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A005515 Number of n-bead bracelets (turn over necklaces) with 10 red 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 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).

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.

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*Plus@@(EulerPhi[#]*(1-x^#)^(-(k/#))&/@Divisors[k]) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2. (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

AUTHOR

N. J. A. Sloane

EXTENSIONS

Sequence extended and description corrected by Christian G. Bower

STATUS

approved

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Last modified October 16 12:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)