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A005516 Number of n-bead bracelets (turn over necklaces) with 12 red beads.
(Formerly M4368)
2
1, 1, 7, 19, 72, 196, 561, 1368, 3260, 7105, 14938, 29624, 56822, 104468, 186616, 322786, 544802, 896259, 1444147, 2278640, 3532144, 5380034, 8070400, 11926928, 17393969, 25042836, 35638596, 50152013, 69855536 (list; graph; refs; listen; history; text; internal format)
OFFSET

12,3

COMMENTS

From Vladimir Shevelev, Apr 23 2011: (Start)

Also number of non-equivalent necklaces of 12 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=12 (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=12..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

Let s(n,k,d)=1, if n==k (mod d), s(n,k,d)=0, otherwise. Then a(n)=s(n,0,12)/6+(n-6)*s(n,0,6)/72+(n-4)*(n-8)*s(n,0,4)/384+(n-3)*(n-6)*(n-9)*s(n,0,3)/1944+(3840*C(n-1,11)+(n+1)*(n-2)*(n-4)*(n-6)*(n-8)*(n-10))/92160, if n is even; a(n)=(n-3)*(n-6)*(n-9)*s(n,0,3)/1944+(3840*C(n-1,11)+(n-1)*(n-3)*(n-5)*(n-7)*(n-9)*(n-11))/92160, if n is odd. - Vladimir Shevelev, Apr 23 2011

From Herbert Kociemba, Nov 04 2016: (Start)

G.f.: 1/2*x^12*((1+x)/(1-x^2)^7 + 1/12*(1/(-1+x)^12 + 1/(-1+x^2)^6 + 2/(-1+x^3)^4 - 2/(-1+x^4)^3 + 2/(-1+x^6)^2 - 4/(-1+x^12))).

G.f.: k=12, x^k*(1/k*Plus@@(EulerPhi[#]*(1-x^#)^(-(k/#))&/@Divisors[k]) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2.

(End)

MATHEMATICA

k = 12; 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=12; CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[k/2+1])/2, {x, 0, 50}], x] (* Herbert Kociemba, Nov 04 2016 *)

CROSSREFS

Sequence in context: A155333 A155296 A155463 * A152008 A002533 A111011

Adjacent sequences:  A005513 A005514 A005515 * A005517 A005518 A005519

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 February 25 21:11 EST 2018. Contains 299657 sequences. (Running on oeis4.)