Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #7 Feb 01 2015 09:39:34
%S 29,7316,285400,4508716,42071257,273611708,1379620392,5723597124,
%T 20398039209,64302648044,183316772048,480140522044,1170651602665
%N a(n) = (1/2)*number of non-degenerate triangular pyramids that can be formed using 4 distinct points chosen from an (n+1) X (n+1) X (n+1) lattice cube.
%C The observed growth rate of CPU time required to compute more terms is approximately ~ n^10.5.
%e a(1)=29: Only 58 of the A103157(1)=70 possible ways to choose 4 distinct points from the 8 vertices of a cube result in pyramids with volume > 0: 2 regular tetrahedra of volume=1/3 and 56 triangular pyramids of volume=1/6. The remaining A103658(1)=12 configurations result in objects with volume=0. Therefore a(1)=(1/2)*(A103157(1)-A103658(1))=58/2=29.
%Y Cf. A103157 binomial((n+1)^3, 4), A103158 tetrahedra in lattice cube, A103658 4-point objects with volume=0 in lattice cube, A103426 non-degenerate triangles in lattice cube.
%K hard,more,nonn
%O 1,1
%A _Hugo Pfoertner_, Feb 14 2005