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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
29, 7316, 285400, 4508716, 42071257, 273611708, 1379620392, 5723597124, 20398039209, 64302648044, 183316772048, 480140522044, 1170651602665 (list; graph; refs; listen; history; text; internal format)



The observed growth rate of CPU time required to compute more terms is approximately ~ n^10.5.


Table of n, a(n) for n=1..13.


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.


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.

Sequence in context: A267955 A267909 A265464 * A201489 A028459 A199369

Adjacent sequences:  A103653 A103654 A103655 * A103657 A103658 A103659




Hugo Pfoertner, Feb 14 2005



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Last modified December 7 09:33 EST 2019. Contains 329843 sequences. (Running on oeis4.)