[seqfan] The number of orbits of triples of (1, 2, ..., n) under the action of the dihedral group of order 2n

[W. Edwin Clark]

I just submitted this sequence (A236283) which I was surprised to not find
in the OEIS. It is an analogue of a question that arose for the action of
another group.  It doesn't seem to fit any of the necklace sequences.
Perhaps someone can think of a nice geometric interpretation.

The number of orbits of triples of (1,2,...,n) under the action of the
dihedral group of order 2n

1, 4, 5, 10, 13, 20, 25, 34, 41, 52, 61, 74, 85, 100, 113, 130, 145, 164,
181, 202, 221, 244, 265, 290, 313, 340, 365, 394, 421, 452, 481, 514, 545,
580, 613, 650, 685, 724, 761, 802, 841, 884, 925, 970, 1013, 1060, 1105,
1154, 1201, 1252

EXAMPLE
For n = 3 there are 5 orbits of triples:
[ [ 1, 1, 1 ], [ 2, 2, 2 ], [ 3, 3, 3 ] ]
[ [ 1, 1, 2 ], [ 2, 2, 3 ], [ 1, 1, 3 ], [ 3, 3, 1 ], [ 3, 3, 2 ], [ 2, 2,
1 ] ]
[ [ 1, 2, 1 ], [ 2, 3, 2 ], [ 1, 3, 1 ], [ 3, 1, 3 ], [ 3, 2, 3 ], [ 2, 1,
2 ] ]
[ [ 1, 2, 2 ], [ 2, 3, 3 ], [ 1, 3, 3 ], [ 3, 1, 1 ], [ 3, 2, 2 ], [ 2, 1,
1 ] ]
[ [ 1, 2, 3 ], [ 2, 3, 1 ], [ 1, 3, 2 ], [ 3, 1, 2 ], [ 3, 2, 1 ], [ 2, 1,
3 ] ]


(GAP)
a:=function(n)
local g, orbs;
g:=DihedralGroup(IsPermGroup, 2*n);
orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 3), OnTuples );
return Size(orbs);
end;;

Maple's guessgf gives this generation function:
-(2*x^3-3*x^2+2*x+1)/(x^4-2*x^3+2*x-1)

--Edwin