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A219965 Number of non-intersecting unit cubes regularly packed into the tetrahedron of edge length n. 1
0, 0, 0, 1, 4, 10, 19, 30, 45, 66, 94, 130, 172, 221, 278, 344, 422, 511, 611, 723, 848, 987, 1140, 1308, 1491, 1691, 1909, 2146, 2401, 2673, 2965, 3278, 3614, 3974, 4355, 4759, 5186, 5638, 6117, 6623, 7156, 7716, 8305, 8923, 9571, 10249, 10958, 11700, 12475, 13285, 14127, 15003, 15914, 16862, 17849, 18874, 19937, 21037, 22177, 23358, 24581, 25846, 27153, 28504 (list; graph; refs; listen; history; text; internal format)



The tetrahedron may be aligned with the Cartesian axes by putting its triangular basis on the horizontal plane, with four vertices at (x, y, z) = (0, 0, 0), (n, 0, 0), (n/2, sqrt(3)*n/2, 0) and (n/2, n/(2*sqrt(3)), n*sqrt(2/3)) see A194082, A020769, A157697.

The volume of tetrahedron is a third times the area of the base triangle times height, (1/3) * (sqrt(3)*n^2/4) * n*sqrt(2/3) = n^3/(3*2^(2/3) = A020829*n^3. This defines an obvious upper limit of floor(n^3/sqrt(72)) = A171973(n) of placing unit cubes into this tetrahedron.

Regular packing: We place the first layer of unit cubes so they touch the floor of the tetrahedron. Their number is limited by the area of the triangular horizontal section of the plane z=1 inside the tetrahedron, which touches all of them; this isosceles horizontal triangle has edge length E(n,z) = n-z*sqrt(3/2). This edge length is a linear interpolation for triangular horizontal cuts between z=0 at the bottom and the summit of the tetrahedron at z=n*sqrt(2/3).

This first layer confined by a triangle characterized by E(n,z) may host RegSquInTri(E) := sum_{y=1..floor(E*sqrt(3)/2)} floor(E-y*2/sqrt(3)) cubes, following recursively the same regular placement and counting strategy as for squares in isosceles triangles, see A194082.

The number of unit cubes in the next layer, between z=1 and z=2, is limited by the area of the horizontal section of the triangle z=2 inside the tetrahedron, where the triangle has edge length n-z*sqrt(3/2).

So in layer z=1, 2, ... we insert ReqSquInTri(E(n,z)) cubes. a(n) is the sum over all these layers with z limited by the z-value of the vertex at the summit.

There is a generalization to placing unit cubes of higher dimensions into higher dimensional tetrahedra.

The growth is expected to be roughly equal to the growth of A000292.


Table of n, a(n) for n=1..64.

R. J. Mathar, Illustration of 30 cubes at n=8

R. J. Mathar, Illustration of 66 cubes at n=10

R. J. Mathar, Illustration of 221 cubes at n=14


a(n) <= A171973(n).


# Number or squares in isosceles triangle of edge length n.

RegSquInTri := proc(n)

add(floor(n-2*y/sqrt(3)), y=1..floor(n*sqrt(3)/2)) ;

end proc:

A219965 := proc(n)

local a, z, triedg ;

a := 0 ;

for z from 1 to floor(n*sqrt(2/3)) do

triedg := n-z*sqrt(3/2) ;

a := a+ RegSquInTri(triedg) ;

end do:

return a;

end proc:


Sequence in context: A022785 A241239 A057312 * A008038 A301248 A160425

Adjacent sequences: A219962 A219963 A219964 * A219966 A219967 A219968




R. J. Mathar, Dec 02 2012



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Last modified December 4 20:58 EST 2022. Contains 358570 sequences. (Running on oeis4.)