cp's OEIS Frontend

A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.

%I A117609 #40 Aug 03 2025 09:00:23
%S A117609 1,7,19,27,33,57,81,81,93,123,147,171,179,203,251,251,257,305,341,365,
%T A117609 389,437,461,461,485,515,587,619,619,691,739,739,751,799,847,895,925,
%U A117609 949,1021,1021,1045,1141,1189,1213,1237,1309,1357,1357,1365,1419,1503
%N A117609 Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.
%H A117609 T. D. Noe, <a href="/A117609/b117609.txt">Table of n, a(n) for n = 0..1000</a>
%H A117609 S. K. K. Choi, A. V. Kumchev and R. Osburn, <a href="https://arxiv.org/abs/math/0502007">On sums of three squares</a>, arXiv:math/0502007 [math.NT], 2005.
%F A117609 a(n) ~ (4/3)*Pi*n^(3/2) ~ A210639(n).
%F A117609 a(n) = A122510(3,n). - _R. J. Mathar_, Apr 21 2010
%F A117609 G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*Sum_{k>=1} q^(k^2). - _Joerg Arndt_, Apr 08 2013
%F A117609 a(n^2) = A000605(n). - _R. J. Mathar_, Aug 03 2025
%e A117609 a(2) = 1 + 6 + 12 = 19, since (0,0,0) and (0, 0, +-1) and cyclic permutations (for a total of 6 points), and +-(0, 1, +-1) and cyclic permutations (for a total 12 points) are inside or on x^2 + y^2 + z^2 = 2.
%t A117609 Table[Sum[SquaresR[3,k], {k,0,n}], {n,0,50}] (* _T. D. Noe_, Apr 08 2006, revised Sep 27 2011 *)
%o A117609 (PARI) A117609(n)=sum(x=0,sqrtint(n),(sum(y=1,sqrtint(t=n-x^2),1+2*sqrtint(t-y^2))*2+sqrtint(t)*2+1)*2^(x>0)) \\ _M. F. Hasler_, Mar 26 2012
%o A117609 (PARI) q='q+O('q^66); Vec((eta(q^2)^5/(eta(q)^2*eta(q^4)^2))^3/(1-q)) /* _Joerg Arndt_, Apr 08 2013 */
%o A117609 (Python)
%o A117609 # uses Python code for A057655
%o A117609 from math import isqrt
%o A117609 def A117609(n): return A057655(n)+(sum(A057655(n-k**2) for k in range(1,isqrt(n)+1))<<1) # _Chai Wah Wu_, Jun 23 2024
%Y A117609 Partial sums of A005875.
%Y A117609 Cf. A000605 (number of points of norm <= n in cubic lattice).
%Y A117609 Cf. A210639, A000092 and references therein.
%Y A117609 Cf. A057655.
%K A117609 nonn
%O A117609 0,2
%A A117609 _John L. Drost_, Apr 06 2006