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A005514 Number of n-bead bracelets (turn over necklaces) with 8 red beads.
(Formerly M3801)
4
1, 1, 5, 10, 29, 57, 126, 232, 440, 750, 1282, 2052, 3260, 4950, 7440, 10824, 15581, 21879, 30415, 41470, 56021, 74503, 98254, 127920, 165288, 211276, 268228, 337416, 421856, 523260, 645456, 790704, 963793, 1167645, 1408185 (list; graph; refs; listen; history; text; internal format)
OFFSET

8,3

COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 8 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=8 (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=8..42.

S. J. Cyvin et al., Polygonal systems including the corannulene and coronene homologs: novel applications of PĆ³lya's theorem, Z. Naturforsch., 52a (1997), 867-873.

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

S. J. Cyvin et al. give a g.f.

G.f.: x^8/16*(1/(1 - x)^8 + 4/(1 - x^8) + 5/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x)^2/(1 - x^2)^3) = x^8*(2*x^10 - 3*x^9 + 7*x^8 - 6*x^7 + 7*x^6 - 2*x^5 + 2*x^4 - 2*x^3 + 5*x^2 - 3*x + 1)/(1 - x)^8/(1 + x)^4/(1 + x^2)^2/(1 + x^4). - Vladeta Jovovic, Jul 17 2002

From Vladimir Shevelev, Apr 23 2011: (Start)

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

a(n) = ((n+4)/32)*s(n,0,8)+((n-4)/32)*s(n,4,8)+(48*C(n-1,7)+(n+1)*(n-2)*(n-4)*(n-6))/768, if n is even; a(n) = (48*C(n-1,7)+(n-1)*(n-3)*(n-5)*(n-7))/768, if n odd.

(End)

G.f.: k=8, x^k*(1/k*Plus@@(EulerPhi[#]*(1-x^#)^(-(k/#))&/@Divisors[k]) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2. - Herbert Kociemba, Nov 05 2016

MATHEMATICA

k = 8; 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=8; 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: A105862 A093029 A105505 * A069921 A053818 A294286

Adjacent sequences:  A005511 A005512 A005513 * A005515 A005516 A005517

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Sequence extended and description corrected by Christian G. Bower

STATUS

approved

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Last modified February 22 23:32 EST 2018. Contains 299472 sequences. (Running on oeis4.)