A014465 A063691 without zeros.

1, 3, 3, 3, 1, 6, 3, 3, 3, 6, 3, 3, 6, 4, 6, 6, 6, 3, 6, 3, 9, 9, 6, 3, 3, 6, 6, 1, 6, 6, 6, 6, 12, 6, 6, 9, 6, 12, 6, 12, 3, 3, 12, 6, 3, 3, 12, 7, 3, 12, 6, 12, 3, 9, 6, 15, 3, 15, 12, 6, 6, 12, 3, 3, 12, 9, 18, 6, 6, 12, 6, 9, 4, 6, 18, 9, 12, 6, 6, 12, 9, 6, 9, 12, 6, 12, 18, 18, 15, 6, 6, 21, 3, 9, 12, 9, 6, 12

Offset

1,2

Comments

Let b(n) = n-th number of form x^2 + y^2 + z^2, x,y,z >= 1 (A000408); a(n) = number of solutions (x,y,z) to x^2 + y^2 + z^2 = b(n).

The a(n) are also the degeneracies of the energy levels E(n) in the three-dimensional cubic "particle-in-a-box" model in elementary quantum mechanics. - A. Timothy Royappa, Jan 09 2009

Continuously increase the radius of a sphere centered at the origin. Whenever the number of entire unit cubes that fit into one quadrant of the sphere increases (cf. A237707), list the number of additional cubes. - M. F. Hasler, Jun 25 2022

References

G. M. Barrow, Physical Chemistry (6th ed.), McGraw-Hill, 1996, p. 69.

Links

Daniel Leary, Table of n, a(n) for n = 1..8283

Example

b(1) = 3 = 1^2 + 1^2 + 1^2 (1 way), so a(1) = 1;
b(2) = 6 = 2^2 + 1^2 + 1^2 (3 ways), so a(2) = 3; etc.

Prog

(PARI) for(n=1, 200, r=sqrtint(n); s=0; for(i=1, r, si=i*i; for(j=1, r, sj=j*j; for(k=1, r, if(si+sj+k*k==n, s=s+1)))); if(s, print1(s, ", "))) /* Ralf Stephan, Aug 31 2013 */

Crossrefs

First différences of A237707.

Sequence in context: A163644 A339901 A290348 * A226645 A243095 A304586

Adjacent sequences: A014462 A014463 A014464 * A014466 A014467 A014468

Keyword

nonn,easy

Author

A. Timothy Royappa, 1997; entry revised Jun 13 2003

Extensions

More terms and better name from Ralf Stephan, Aug 31 2013

Status

approved