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

1,2

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

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).

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

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]].

Prog

(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

Crossrefs

Cf. A236332 (4-tuples).

Sequence in context: A094415 A114517 A283246 * A378273 A357865 A366864

Adjacent sequences: A236280 A236281 A236282 * A236284 A236285 A236286

Keyword

nonn,easy

Author

W. Edwin Clark, Jan 21 2014

Status

approved