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%I #30 Jun 25 2022 11:27:21
%S 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,
%T 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,
%U 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
%N A063691 without zeros.
%C 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).
%C 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
%C 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
%D G. M. Barrow, Physical Chemistry (6th ed.), McGraw-Hill, 1996, p. 69.
%H Daniel Leary, <a href="/A014465/b014465.txt">Table of n, a(n) for n = 1..8283</a>
%e b(1) = 3 = 1^2 + 1^2 + 1^2 (1 way), so a(1) = 1;
%e b(2) = 6 = 2^2 + 1^2 + 1^2 (3 ways), so a(2) = 3; etc.
%o (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 */
%Y First différences of A237707.
%K nonn,easy
%O 1,2
%A _A. Timothy Royappa_, 1997; entry revised Jun 13 2003
%E More terms and better name from _Ralf Stephan_, Aug 31 2013
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