OFFSET
2,1
COMMENTS
The sequence is inspired by a problem in astronomy, where it is useful for obtaining the volume of global subsurface oceans of moons or planets.
The volume of a sphere of radius r is (4/3)*Pi*r^3. The volume of the spherical shell formed by two concentric spheres is then obtained by subtracting the volume of the smaller sphere from the volume of the larger sphere.
There is evidence that Saturn's moon Enceladus contains a global subsurface ocean of liquid water. It is suspected that the top of the ocean lies beneath a 30 to 40 kilometer thick ice shelf. Measurements of Enceladus's libration suggest that the ocean is about 26 to 31 kilometers deep (see Enceladus, Wikipedia). Enceladus has a radius of approximately 252 kilometers. If the thickness of the ice shelf is assumed to be at the upper bound of 40 kilometers and the ocean being 26 kilometers deep, this would give n = 212 for the outer radius of the ocean and k = 186 for the inner radius of the ocean. T(212, 186) = 12957067, so Enceladus's global subsurface ocean should contain at least about 12957067 cubic kilometers of water, which is about 1 percent of Earth's World Ocean of approximately 1332000000 cubic kilometers.
The exact values of the volumes obtained for solar system bodies are not integers. However, the difference between the rounded value and the non-rounded value is often insignificant for useful comparisons like the one given in the previous comment.
LINKS
EXAMPLE
The triangle starts as follows:
29;
109, 80;
264, 235, 155;
519, 490, 411, 256;
901, 871, 792, 637, 381;
1433, 1403, 1324, 1169, 913, 532;
2140, 2111, 2032, 1877, 1621, 1240, 708;
3049, 3020, 2941, 2786, 2530, 2149, 1617, 909;
4185, 4155, 4076, 3921, 3665, 3284, 2752, 2044, 1135;
5571, 5542, 5462, 5307, 5052, 4671, 4139, 3431, 2522, 1386;
PROG
(PARI) shellvolume(n, k) = (4/3)*Pi*(n^3-k^3)
T(n, k) = round(shellvolume(n, k))
trianglerows(n) = for(x=2, n+1, for(y=1, x-1, print1(T(x, y), ", ")); print(""))
trianglerows(10) \\ print initial 10 rows of triangle
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Felix Fröhlich, Sep 19 2016
STATUS
approved