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A224215 Number of nonnegative solutions to x^3 + y^3 + z^3 <= n^3.

%I A224215 #35 Feb 01 2025 14:57:21
%S A224215 1,4,11,30,66,115,200,302,441,619,829,1085,1395,1771,2200,2666,3228,
%T A224215 3843,4564,5351,6185,7143,8158,9349,10526,11934,13375,14896,16652,
%U A224215 18381,20370,22411,24629,26963,29406,32101,34840,37766,40920,44164,47587,51200
%N A224215 Number of nonnegative solutions to x^3 + y^3 + z^3 <= n^3.
%H A224215 David A. Corneth, <a href="/A224215/b224215.txt">Table of n, a(n) for n = 0..1500</a>
%H A224215 David A. Corneth, <a href="/A224215/a224215_1.gp.txt">Pari prog</a>
%F A224215 a(n) = [x^(n^3)] (1/(1 - x))*(Sum_{k>=0} x^(k^3))^3. - _Ilya Gutkovskiy_, Apr 20 2018
%e A224215 For n=1, the four solutions are {0,0,0}, {0,0,1}, {0,1,0} and {1,0,0}, so a(1)=4.
%o A224215 (Python)
%o A224215 for a in range(99):
%o A224215   n = a*a*a
%o A224215   k = 0
%o A224215   for x in range(99):
%o A224215     s = x*x*x
%o A224215     if s>n: break
%o A224215     for y in range(99):
%o A224215         sy = s + y*y*y
%o A224215         if sy>n: break
%o A224215         for z in range(99):
%o A224215             sz = sy + z*z*z
%o A224215             if sz>n: break
%o A224215             k+=1
%o A224215   print(k, end=',')
%o A224215 (PARI) a(n) = n++; p = Pol((1/(1 - x))*sum(k=0, n, x^(k^3))^3 + O(x^(n^3))); polcoeff(p, (n-1)^3); \\ _Michel Marcus_, Apr 21 2018
%o A224215 (PARI) \\ See PARI link. _David A. Corneth_, May 22 2018
%Y A224215 Cf. A224214.
%K A224215 nonn
%O A224215 0,2
%A A224215 _Alex Ratushnyak_, Apr 01 2013