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

Author

Alford Arnold

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A006565.

Programs

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

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 *)

PARI

a(n)=n*(n+1)*(n^4-n^3+4*n^2+2)/12 \\ Charles R Greathouse IV, Feb 24 2011

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

Extensions

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