A027670 Number of different bracelets with 6 beads of at most n colors, allowing turning over.

0, 1, 13, 92, 430, 1505, 4291, 10528, 23052, 46185, 86185, 151756, 254618, 410137, 638015, 963040, 1415896, 2034033, 2862597, 3955420, 5376070, 7198961, 9510523, 12410432, 16012900, 20448025, 25863201

Offset

0,3

Comments

Number of ways to color vertices of a hexagon using <= n colors, allowing rotations and reflections.

Equivalently, the number of distinct hexagons that can be tiled using equilateral triangles of n different colors. - Lekraj Beedassy, Dec 29 2007

Number of ways to color slots of a 2 X 3 matrix with the respective symmetric groups S_2 and S_3 acting on the rows / columns. - Marko Riedel, Jan 26 2017

References

J. L. Fisher, Application-Oriented Algebra (1977), ISBN 0-7002-2504-8, circa p. 215.

M. Gardner, New Mathematical Diversions from Scientific American, Simon and Schuster, New York, 1966, pages 245-246.

J.-P. Delahaye, Le miraculeux "lemme de Burnside"; Groupes et orbites, pp. 146-147, in 'Pour la Science' (French edition of 'Scientific American'), No. 350, December 2006, Paris.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Marko Riedel, Worked example for computing the cycle index for the shuffled two-by-three matrix.

Index entries for sequences related to bracelets

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

1/12*n*(n+1)*(n^4 - n^3 + 4*n^2 + 2).

G.f.: x*(1+x)*(1 + 5*x + 17*x^2 + 7*x^3)/(1-x)^7. - Colin Barker, Jan 29 2012

Cycle index: s1^6/12 + s2^3/3 + s3^2/6 + s1^2 * s2^2/4 + s6/6, -Marko Riedel, Jan 26 2017

Maple

A027670 := n-> (n^6+3*n^4+4*n^3+2*n^2+2*n)/12;

Mathematica

(* First do *) Needs["Combinatorica`"] (* then *) Table[ CycleIndex[ DihedralGroup[6], t] /. Table[ t[i] -> n, {i, 1, 6}], {n, 0, 26}]
CoefficientList[Series[x*(1+x)*(1+5*x+17*x^2+7*x^3)/(1-x)^7, {x, 0, 30}], x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{7, -21, 35, -35, 21, -7, 1}, {0, 1, 13, 92, 430, 1505, 4291}, 30] (* Harvey P. Dale, Mar 12 2018 *)

Prog

(PARI) a(n)=n*(n+1)*(n^4-n^3+4*n^2+2)/12 \\ Charles R Greathouse IV, Feb 24 2011
(Magma) I:=[0, 1, 13, 92, 430, 1505, 4291]; [n le 7 select I[n] else 7*Self(n-1)-21*Self(n-2)+35*Self(n-3)-35*Self(n-4)+21*Self(n-5)-7*Self(n-6)+Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 22 2012

Crossrefs

Cf. A006565.

Sequence in context: A092469 A300779 A275918 * A055608 A038742 A282709

Adjacent sequences: A027667 A027668 A027669 * A027671 A027672 A027673

Keyword

nonn,easy

Author

Alford Arnold

Extensions

Name changed to reflect distinction between necklaces (cyclic) and bracelets (dihedral) by Marko Riedel, Jan 27 2017

Status

approved