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A249460 Number of proper colorings of the cube with at most n colors under rotational symmetry.

%I A249460 #33 Sep 08 2022 08:46:10
%S A249460 0,0,0,1,10,55,230,770,2156,5250,11460,22935,42790,75361,126490,
%T A249460 203840,317240,479060,704616,1012605,1425570,1970395,2678830,3588046,
%U A249460 4741220,6188150,7985900,10199475,12902526,16178085,20119330
%N A249460 Number of proper colorings of the cube with at most n colors under rotational symmetry.
%C A249460 All terms 3 mod 10 end in 1, all terms 8 mod 10 end in 6 and vice versa. - _Jon Perry_, Oct 29 2014
%C A249460 Also the number of inequivalent ways to color the corners of an octahedron using at most n colors so that no two adjacent corners have the same color. - _David Nacin_, Feb 22 2017
%H A249460 Marko R. Riedel, <a href="http://math.stackexchange.com/questions/994998/">Proper colorings of the cube</a>, Mathematics Stack Exchange, Oct 29 2014
%H A249460 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A249460 a(n) = ( n*(n-1)*(n-2)*(n^3-9*n^2+29*n-32) + 3*n*(n-1)*(n-2)^2 )/24 = (n-2)*(n-1)*n*(n^3-9*n^2+32*n-38)/24.
%F A249460 G.f.: x^3*(1+3*x+6*x^2+20*x^3)/(1-x)^7. - _Vincenzo Librandi_, Oct 30 2014
%e A249460 For n = 3 we see there is only a(3) = 1 way to color the faces of a cube with three colors so that no two adjacent sides have the same color. - _David Nacin_, Feb 22 2017
%p A249460 q := N -> 1/24*(N*(N-1)*(N-2)*(N^3-9*N^2+29*N-32) + 3*N*(N-1)*(N-2)^2);
%t A249460 Table[(n - 2) (n - 1) n (n^3 - 9 n^2 + 32 n - 38)/24, {n, 0, 30}] (* _Bruno Berselli_, Oct 30 2014 *)
%t A249460 CoefficientList[Series[x^3 (1 + 3 x + 6 x^2 + 20 x^3)/(1 - x)^7, {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 30 2014 *)
%o A249460 (Magma) [(n-2)*(n-1)*n*(n^3-9*n^2+32*n-38)/24: n in [0..30]]; // _Vincenzo Librandi_, Oct 30 2014
%o A249460 (PARI) a(n)=(n-2)*(n-1)*n*(n^3-9*n^2+32*n-38)/24 \\ _Charles R Greathouse IV_, Feb 23 2017
%K A249460 nonn,easy
%O A249460 0,5
%A A249460 _Marko Riedel_, Oct 29 2014