A236332 The number of orbits of 4-tuples of the dihedral group of order 2n acting on {1,2,...,n}.

1, 8, 14, 36, 63, 112, 172, 260, 365, 504, 666, 868, 1099, 1376, 1688, 2052, 2457, 2920, 3430, 4004, 4631, 5328, 6084, 6916, 7813, 8792, 9842, 10980, 12195, 13504, 14896, 16388, 17969, 19656, 21438, 23332, 25327, 27440, 29660, 32004, 34461, 37048, 39754, 42596, 45563

Offset: 1

Author

W. Edwin Clark, Jan 22 2014

Keywords

• nonn
• easy

Comments

In other words, a(n) is the number of equivalence classes of length 4 words with an alphabet of size n where equivalence is up to rotation or reflection of the alphabet. - Andrew Howroyd, Jan 17 2020

Examples

			For n = 3 there are the following 14 orbits of 4-tuples for the group D6 = S3:
1) [[1,1,1,1], [2,2,2,2], [3,3,3,3]],
2) [[1,1,1,2], [2,2,2,3], [1,1,1,3], [3,3,3,1], [3,3,3,2], [2,2,2,1]],
3) [[1,1,2,1], [2,2,3,2], [1,1,3,1], [3,3,1,3], [3,3,2,3], [2,2,1,2]],
4) [[1,1,2,2], [2,2,3,3], [1,1,3,3], [3,3,1,1], [3,3,2,2], [2,2,1,1]],
5) [[1,1,2,3], [2,2,3,1], [1,1,3,2], [3,3,1,2], [3,3,2,1], [2,2,1,3]],
6) [[1,2,1,1], [2,3,2,2], [1,3,1,1], [3,1,3,3], [3,2,3,3], [2,1,2,2]],
7) [[1,2,1,2], [2,3,2,3], [1,3,1,3], [3,1,3,1], [3,2,3,2], [2,1,2,1]],
8) [[1,2,1,3], [2,3,2,1], [1,3,1,2], [3,1,3,2], [3,2,3,1], [2,1,2,3]],
9) [[1,2,2,1], [2,3,3,2], [1,3,3,1], [3,1,1,3], [3,2,2,3], [2,1,1,2]],
10) [[1,2,2,2], [2,3,3,3], [1,3,3,3], [3,1,1,1], [3,2,2,2], [2,1,1,1]],
11) [[1,2,2,3], [2,3,3,1], [1,3,3,2], [3,1,1,2], [3,2,2,1], [2,1,1,3]],
12) [[1,2,3,1], [2,3,1,2], [1,3,2,1], [3,1,2,3], [3,2,1,3], [2,1,3,2]],
13) [[1,2,3,2], [2,3,1,3], [1,3,2,3], [3,1,2,1], [3,2,1,2], [2,1,3,1]],
14) [[1,2,3,3], [2,3,1,1], [1,3,2,2], [3,1,2,2], [3,2,1,1], [2,1,3,3]].
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A236283 (3-tuples).

Programs

GAP

a:=function(n)
local g,orbs;
g:=DihedralGroup(IsPermGroup,2*n);
orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 4), OnTuples );
return Size(orbs);
end;;

PARI

a(n) = {(9 + 7*(-1)^n + 2*n^3)/4} \\ Andrew Howroyd, Jan 17 2020

Formula

Conjectures from Colin Barker, Jan 22 2014: (Start)

a(n) = (9 + 7*(-1)^n + 2*n^3)/4.

G.f.: -x*(4*x^4-12*x^3+8*x^2-5*x-1) / ((x-1)^4*(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*n + 4)/2 + 8 for even n.

a(n) = (n-1)*(n^2 + n + 1)/2 + 1 for odd n.

(End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 17 2020