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A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2.

%I A253663 #31 Aug 03 2025 09:03:49
%S A253663 0,0,1,7,17,38,78,127,196,296,410,564,738,958,1220,1514,1848,2235,
%T A253663 2686,3175,3719,4365,5007,5758,6568,7442,8415,9477,10597,11779,13100,
%U A253663 14459,15954,17566,19231,21029,22916,24930,27030,29293,31616,34103,36732,39459
%N A253663 Number of positive solutions to x^2+y^2+z^2 <= n^2.
%C A253663 Whereas A000604 counts solutions where x>=0, y>=0, z>=0, this sequence counts solutions where x>0, y>0, z>0.
%H A253663 Alois P. Heinz, <a href="/A253663/b253663.txt">Table of n, a(n) for n = 0..1000</a>
%F A253663 a(n) = A211639(n^2).
%F A253663 a(n) = [x^(n^2)] (theta_3(x) - 1)^3/(8*(1 - x)), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Apr 17 2018
%F A253663 Comment from _N. J. A. Sloane_, Jun 02 2024 (Start)
%F A253663 The one-dimensional lattice {n: n an integer} , which graphically looks like
%F A253663    ...o   o   o   o   o   o   ...
%F A253663 has theta series 1 + 2 q + 2 q^4 + 2 q^9 + 2 q^16 + ... = sum {n=-oo..oo} q^(n^2),
%F A253663 and that power series is called theta_3(q), A000122.
%F A253663   Raising it to the power 3 counts points with x^2+y^2+z^2 = k, A005875.
%F A253663 Dividing it by 1-x gives the partial sums, which basically is what this sequence is.
%F A253663 So a first approximation to a theta series for the sequence is theta_3(q)^8/(1-q).
%F A253663 Subtracting 1 and dividing by 8 is because here we only want positive solutions.
%F A253663 (End)
%e A253663 a(4)=17 counts the following solutions (x,y,z): (1,1,1), (2,2,2), three permutations of (1,1,2), three permutations of (1,1,3), three permutations of (1,2,2), and six permutations of (1,2,3).
%o A253663 (Sage)
%o A253663 [len([(x,y,z) for x in [1..n] for y in [1..n] for z in [1..n] if x^2+y^2+z^2<=n^2]) for n in [0..43]] # _Tom Edgar_, Jan 07 2015
%Y A253663 Cf. A000604.
%K A253663 nonn
%O A253663 0,4
%A A253663 _R. J. Mathar_, Jan 07 2015