%I #51 Sep 08 2022 08:44:49
%S 0,1,13,92,430,1505,4291,10528,23052,46185,86185,151756,254618,410137,
%T 638015,963040,1415896,2034033,2862597,3955420,5376070,7198961,
%U 9510523,12410432,16012900,20448025,25863201
%N Number of different bracelets with 6 beads of at most n colors, allowing turning over.
%C Number of ways to color vertices of a hexagon using <= n colors, allowing rotations and reflections.
%C Equivalently, the number of distinct hexagons that can be tiled using equilateral triangles of n different colors. - _Lekraj Beedassy_, Dec 29 2007
%C 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
%D J. L. Fisher, Application-Oriented Algebra (1977), ISBN 0-7002-2504-8, circa p. 215.
%D M. Gardner, New Mathematical Diversions from Scientific American, Simon and Schuster, New York, 1966, pages 245-246.
%D 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.
%H Vincenzo Librandi, <a href="/A027670/b027670.txt">Table of n, a(n) for n = 0..1000</a>
%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%H Marko Riedel, <a href="http://math.stackexchange.com/questions/2113657/">Worked example for computing the cycle index for the shuffled two-by-three matrix.</a>
%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F 1/12*n*(n+1)*(n^4 - n^3 + 4*n^2 + 2).
%F G.f.: x*(1+x)*(1 + 5*x + 17*x^2 + 7*x^3)/(1-x)^7. - _Colin Barker_, Jan 29 2012
%F Cycle index: s1^6/12 + s2^3/3 + s3^2/6 + s1^2 * s2^2/4 + s6/6, -_Marko Riedel_, Jan 26 2017
%p A027670 := n-> (n^6+3*n^4+4*n^3+2*n^2+2*n)/12;
%t (* First do *) Needs["Combinatorica`"] (* then *) Table[ CycleIndex[ DihedralGroup[6], t] /. Table[ t[i] -> n, {i, 1, 6}], {n, 0, 26}]
%t 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 *)
%t LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,13,92,430,1505,4291},30] (* _Harvey P. Dale_, Mar 12 2018 *)
%o (PARI) a(n)=n*(n+1)*(n^4-n^3+4*n^2+2)/12 \\ _Charles R Greathouse IV_, Feb 24 2011
%o (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
%Y Cf. A006565.
%K nonn,easy
%O 0,3
%A _Alford Arnold_
%E Name changed to reflect distinction between necklaces (cyclic) and bracelets (dihedral) by _Marko Riedel_, Jan 27 2017