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

%I A236332 #16 Jan 17 2020 17:21:09
%S A236332 1,8,14,36,63,112,172,260,365,504,666,868,1099,1376,1688,2052,2457,
%T A236332 2920,3430,4004,4631,5328,6084,6916,7813,8792,9842,10980,12195,13504,
%U A236332 14896,16388,17969,19656,21438,23332,25327,27440,29660,32004,34461,37048,39754,42596,45563
%N A236332 The number of orbits of 4-tuples of the dihedral group of order 2n acting on {1,2,...,n}.
%C A236332 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
%H A236332 Andrew Howroyd, <a href="/A236332/b236332.txt">Table of n, a(n) for n = 1..1000</a>
%H A236332 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A236332 Conjectures from _Colin Barker_, Jan 22 2014: (Start)
%F A236332 a(n) = (9 + 7*(-1)^n + 2*n^3)/4.
%F A236332 G.f.: -x*(4*x^4-12*x^3+8*x^2-5*x-1) / ((x-1)^4*(x+1)).
%F A236332 (End)
%F A236332 From _Andrew Howroyd_, Jan 17 2020: (Start)
%F A236332 The above conjectures are true and can be derived from the following formulas for even and odd n.
%F A236332 a(n) = (n-2)*(n^2 + 2*n + 4)/2 + 8 for even n.
%F A236332 a(n) = (n-1)*(n^2 + n + 1)/2 + 1 for odd n.
%F A236332 (End)
%e A236332 For n = 3 there are the following 14 orbits of 4-tuples for the group D6 = S3:
%e A236332 1) [[1,1,1,1], [2,2,2,2], [3,3,3,3]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]],
%e A236332 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]].
%o A236332 (GAP)
%o A236332 a:=function(n)
%o A236332 local g,orbs;
%o A236332 g:=DihedralGroup(IsPermGroup,2*n);
%o A236332 orbs := OrbitsDomain(g, Tuples( [ 1 .. n ], 4), OnTuples );
%o A236332 return Size(orbs);
%o A236332 end;;
%o A236332 (PARI) a(n) = {(9 + 7*(-1)^n + 2*n^3)/4} \\ _Andrew Howroyd_, Jan 17 2020
%Y A236332 Cf. A236283 (3-tuples).
%K A236332 nonn,easy
%O A236332 1,2
%A A236332 _W. Edwin Clark_, Jan 22 2014
%E A236332 Terms a(21) and beyond from _Andrew Howroyd_, Jan 17 2020