A276776 Volume of spherical shell between two spheres of radius n and k, rounded to the nearest integer. Triangular array T(n, k), read by rows.

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

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

Table of n, a(n) for n=2..48.

Wikipedia, Enceladus - Subsurface water ocean.

Wikipedia, Spherical shell.

Wikipedia, World Ocean.

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

Sequence in context: A142176 A115272 A138716 * A142194 A139495 A139983

Adjacent sequences: A276773 A276774 A276775 * A276777 A276778 A276779

Keyword

nonn,tabl

Author

Felix Fröhlich, Sep 19 2016

Status

approved