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A117609 Number of lattice points inside the ball x^2+y^2+z^2 <= n. 20

%I

%S 1,7,19,27,33,57,81,81,93,123,147,171,179,203,251,251,257,305,341,365,

%T 389,437,461,461,485,515,587,619,619,691,739,739,751,799,847,895,925,

%U 949,1021,1021,1045,1141,1189,1213,1237,1309,1357,1357,1365,1419,1503

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

%H T. D. Noe, <a href="/A117609/b117609.txt">Table of n, a(n) for n = 0..1000</a>

%H S. K. K. Choi, A. V. Kumchev and R. Osburn, <a href="http://arXiv.org/abs/math.NT/0502007">On sums of three squares</a>, arXiv:math/0502007 [math.NT], 2005.

%F a(n) ~ (4/3)*Pi*n^1.5 ~ A210639(n).

%F a(n) = A122510(3,n). [_R. J. Mathar_, Apr 21 2010]

%F G.f.: T3(q)^3/(1-q) where T3(q) = 1 + 2*sum(k>=1, q^(k^2) ). [_Joerg Arndt_, Apr 08 2013]

%e a(2) is 19, since (0,0,0)(1 point) (0,0,1) (6 points with all rearrangements and sign assignments) and (0,1,1) (12 points) are inside or on x^2+y^2+z^2=2.

%t Table[Sum[SquaresR[3,k], {k,0,n}], {n,0,50}] (* _T. D. Noe_, Apr 08 2006, revised Sep 27 2011 *)

%o (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 (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 */

%Y Partial sums of A005875.

%Y Cf. A000605 (number of points of norm <= n in cubic lattice).

%Y Cf. A210639, A000092 and references therein.

%K nonn

%O 0,2

%A _John L. Drost_, Apr 06 2006

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Last modified August 5 07:16 EDT 2021. Contains 346463 sequences. (Running on oeis4.)