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A027670 Number of different bracelets with 6 beads of at most n colors, allowing turning over. 6
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 20 20:16 EDT 2019. Contains 323426 sequences. (Running on oeis4.)