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Number of inequivalent ways to color the faces of a cube using at most n colors so that no two opposite sides have the same color.
2

%I #11 Feb 23 2017 04:32:52

%S 0,0,1,11,76,340,1135,3101,7336,15576,30405,55495,95876,158236,251251,

%T 385945,576080,838576,1193961,1666851,2286460,3087140,4108951,5398261,

%U 7008376,9000200,11442925,14414751,18003636,22308076,27437915,33515185,40674976,49066336

%N Number of inequivalent ways to color the faces of a cube using at most n colors so that no two opposite sides have the same color.

%C Also the number of inequivalent ways to color the corners of an octahedron using at most n colors so that no two opposite corners have the same color.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = n*(n-1)*(n^4-2*n^3+n^2+8)/24.

%F G.f.: -x^2*(1+4*x+20*x^2+4*x^3+x^4)/(x-1)^7 . - _R. J. Mathar_, Feb 23 2017

%e For n = 2 we get a(2) = 1 way to color the faces of a cube with two colors so that no two opposite sides have the same color.

%t Table[(8n(n-1) + n^3(n-1)^3) /24, {n, 0, 35}]

%o (PARI) a(n) = n*(n-1)*(n^4-2*n^3+n^2+8)/24 \\ _Charles R Greathouse IV_, Feb 22 2017

%Y Cf. A282817, A047780 (face colorings without restriction).

%K nonn,easy

%O 0,4

%A _David Nacin_, Feb 21 2017