A236283 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

Offset: 1

Author

W. Edwin Clark, Jan 21 2014

Keywords

• nonn
• easy

Comments

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

Examples

			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]].
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Crossrefs

Cf. A236332 (4-tuples).

Programs

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

Formula

Conjectures from Colin Barker, Jan 21 2014: (Start)

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)

From Andrew Howroyd, Jan 17 2020: (Start)

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)

a(n) = A081352(n - 1) - A116940(n - 1). - Miko Labalan, Nov 12 2016