A097513 - OEIS

%I #6 Sep 10 2013 16:36:38

%S 1,1,3,5,10,15,27,38,60,84,122,164,229,298,398,509,658,823,1041,1278,

%T 1582,1917,2331,2786,3343,3948,4676,5471,6408,7428,8622,9912,11406,

%U 13023,14871,16866,19135,21571,24321,27275,30580,34122,38070,42284,46956,51942

%N Number of ways to label the vertices of the octahedron (or faces of the cube) with nonnegative integers summing to n, where labelings that differ only by rotation or reflection are considered the same.

%F G.f.: (q^8-q^7+q^6+q^4+q^2-q+1)/((-1+q)^6*(q+1)^3*(q^2+q+1)^2*(q^2-q+1)*(q^2+1)).

%F a(n) is asymptotically equal to n^5/5760. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 31 2004

%e a(3) = 5 because we can label the faces of the cube with nonnegative integers summing to three in five ways: 3 on one face, 2 on one face and 1 on an adjacent face, 2 on one face and 1 on the opposite face, 1 on three faces sharing a corner, 1 on three faces not sharing a corner.

%p a:= n-> (Matrix([[1, 0$8, -1$2, -3, -5, -10, -15, -27, -38]]).Matrix(17, (i,j)-> if (i=j-1) then 1 elif j=1 then [2, 0, -1, 0, -2, 3, -2, 1, 1, -2, 3, -2, 0, -1, 0, 2, -1][i] else 0 fi)^n)[1,1]; seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 31 2008

%Y Cf. A006381.

%K easy,nonn

%O 0,3

%A Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004