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A236283
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The number of orbits of triples of {1,2,...,n} under the action of the dihedral group of order 2n.
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6
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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
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OFFSET
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1,2
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COMMENTS
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In other words, a(n) is the number of equivalence classes of length 3 words with an alphabet of size n where equivalence is up to rotation or reflection of the alphabet. For example when n is 3, the word 112 is equivalent to 223 and 331 by rotation of the alphabet, and these are equivalent to 332, 221 and 113 by reflection of the alphabet. - Andrew Howroyd, Jan 17 2020
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LINKS
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FORMULA
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a(n) = (5 + 3*(-1)^n + 2*n^2)/4.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
G.f.: -x*(2*x^3-3*x^2+2*x+1) / ((x-1)^3*(x+1)).
(End)
The above conjectures are true and can be derived from the following formulas for even and odd n.
a(n) = (n-2)*(n + 2)/2 + 4 for even n.
a(n) = (n-1)*(n + 1)/2 + 1 for odd n.
(End)
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EXAMPLE
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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]].
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PROG
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(GAP)
a:=function(n)
local g, orbs;
g:=DihedralGroup(IsPermGroup, 2*n);
orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 3), OnTuples );
return Size(orbs);
end;;
(PARI) a(n) = {(5 + 3*(-1)^n + 2*n^2)/4} \\ Andrew Howroyd, Jan 17 2020
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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