A097513 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.

1, 1, 3, 5, 10, 15, 27, 38, 60, 84, 122, 164, 229, 298, 398, 509, 658, 823, 1041, 1278, 1582, 1917, 2331, 2786, 3343, 3948, 4676, 5471, 6408, 7428, 8622, 9912, 11406, 13023, 14871, 16866, 19135, 21571, 24321, 27275, 30580, 34122, 38070, 42284, 46956, 51942

Offset

0,3

Links

Table of n, a(n) for n=0..45.

Formula

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)).

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

Example

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.

Maple

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

Crossrefs

Cf. A006381.

Sequence in context: A254346 A132302 A308872 * A308932 A308997 A045513

Adjacent sequences: A097510 A097511 A097512 * A097514 A097515 A097516

Keyword

easy,nonn

Author

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

Status

approved