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A103656 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. 4

%I

%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

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Last modified September 19 03:31 EDT 2021. Contains 347550 sequences. (Running on oeis4.)