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A027670
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Number of different necklaces with 6 beads of n colors, allowing turning over.
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3
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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; internal format)
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OFFSET
| 0,3
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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 (blekraj(AT)yahoo.com), Dec 29 2007
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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.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
J.-P. Delahaye, 'Le miraculeux "lemme de Burnside"' ; 'Groupes et orbites' pp 146-7 in 'Pour la Science' (French edition of 'Scientific American') No.350 December 2006 Paris.
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LINKS
| Index entries for sequences related to bracelets
Index entries for sequences related to linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
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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]
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MAPLE
| A027670 := n-> (n^6+3*n^4+4*n^3+2*n^2+2*n)/12;
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MATHEMATICA
| (*First do*) Needs["Combinatorica`"] (*then*) Table[ CycleIndex[ DihedralGroup[6], t] /. Table[ t[i] -> n, {i, 1, 6}], {n, 0, 26}]
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PROG
| (PARI) a(n)=n*(n+1)*(n^4-n^3+4*n^2+2)/12 \\ Charles R Greathouse IV, Feb 24, 2011
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CROSSREFS
| Cf. A006565.
Sequence in context: A022578 A090020 A092469 * A055608 A038742 A005414
Adjacent sequences: A027667 A027668 A027669 * A027671 A027672 A027673
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KEYWORD
| nonn,easy
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AUTHOR
| Alford Arnold (Alford1940(AT)aol.com)
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