cp's OEIS Frontend

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.

%I A097513 #10 Mar 07 2025 11:44:40
%S A097513 1,1,3,5,10,15,27,38,60,84,122,164,229,298,398,509,658,823,1041,1278,
%T A097513 1582,1917,2331,2786,3343,3948,4676,5471,6408,7428,8622,9912,11406,
%U A097513 13023,14871,16866,19135,21571,24321,27275,30580,34122,38070,42284,46956,51942
%N 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.
%H A097513 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A097513 <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,0,-2,3,-2,1,1,-2,3,-2,0,-1,0,2,-1).
%F A097513 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 A097513 a(n) is asymptotically equal to n^5/5760. - Isabel C. Lugo (izzycat(AT)gmail.com), Aug 31 2004
%e A097513 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 A097513 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 A097513 Cf. A006381.
%K A097513 easy,nonn
%O A097513 0,3
%A A097513 Isabel C. Lugo (izzycat(AT)gmail.com), Aug 26 2004