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

Author

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

Keywords

• nonn
• easy

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

Examples

			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.
		

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

Crossrefs

First différences of A237707.

Programs

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 */

Extensions

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