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

0, 0, 0, 1, 10, 55, 230, 770, 2156, 5250, 11460, 22935, 42790, 75361, 126490, 203840, 317240, 479060, 704616, 1012605, 1425570, 1970395, 2678830, 3588046, 4741220, 6188150, 7985900, 10199475, 12902526, 16178085, 20119330

Offset

0,5

Comments

All terms 3 mod 10 end in 1, all terms 8 mod 10 end in 6 and vice versa. - Jon Perry, Oct 29 2014

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

Links

Table of n, a(n) for n=0..30.

Marko R. Riedel, Proper colorings of the cube, Mathematics Stack Exchange, Oct 29 2014

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

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.

G.f.: x^3*(1+3*x+6*x^2+20*x^3)/(1-x)^7. - Vincenzo Librandi, Oct 30 2014

Example

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

Maple

q := N -> 1/24*(N*(N-1)*(N-2)*(N^3-9*N^2+29*N-32) + 3*N*(N-1)*(N-2)^2);

Mathematica

Table[(n - 2) (n - 1) n (n^3 - 9 n^2 + 32 n - 38)/24, {n, 0, 30}] (* Bruno Berselli, Oct 30 2014 *)
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 *)

Prog

(Magma) [(n-2)*(n-1)*n*(n^3-9*n^2+32*n-38)/24: n in [0..30]]; // Vincenzo Librandi, Oct 30 2014
(PARI) a(n)=(n-2)*(n-1)*n*(n^3-9*n^2+32*n-38)/24 \\ Charles R Greathouse IV, Feb 23 2017

Crossrefs

Sequence in context: A034241 A341223 A337631 * A022575 A202481 A348663

Adjacent sequences: A249457 A249458 A249459 * A249461 A249462 A249463

Keyword

nonn,easy

Author

Marko Riedel, Oct 29 2014

Status

approved