This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A274274 #27 Jul 14 2016 21:05:26
%S A274274 1,2,2,1,1,2,1,0,2,3,3,1,1,2,1,0,2,3,3,1,1,2,0,0,1,3,4,2,2,2,1,1,2,3,
%T A274274 2,2,2,4,1,0,3,2,2,1,2,3,1,1,1,2,3,2,3,4,1,0,1,1,3,2,1,3,1,1,3,4,4,1,
%U A274274 3,3,0,0,4,5,3,1,2,3,0,1,4
%N A274274 Number of ordered ways to write n as x^3 + y^2 + z^2, where x,y,z are nonnegative integers with y <= z.
%C A274274 Conjecture: Let n be any nonnegative integer.
%C A274274 (i) Either a(n) > 0 or a(n-2) > 0. Also, a(n) > 0 or a(n-6) > 0. Moreover, if n has the form 2^k*(4m+1) with k and m nonnegative integers, then a(n) > 0 except for n = 813, 4404, 6420, 28804.
%C A274274 (ii) Either n or n-3 can be written as x^3 + y^2 + 3*z^2 with x,y,z nonnegative integers.
%C A274274 (iii) For each d = 4, 5, 11, 12, either n or n-d can be written as x^3 + y^2 + 2*z^2 with x,y,z nonnegative integers.
%C A274274 We have verified that a(n) or a(n-2) is positive for every n = 0..2*10^6. Note that for each n = 0,1,2,... either n or n-2 can be written as x^2 + y^2 + z^2 with x,y,z nonnegative integers, which follows immediately from the Gauss-Legendre theorem on sums of three squares.
%H A274274 Zhi-Wei Sun, <a href="/A274274/b274274.txt">Table of n, a(n) for n = 0..10000</a>
%e A274274 a(6) = 1 since 6 = 1^3 + 1^2 + 2^2.
%e A274274 a(14) = 1 since 14 = 1^3 + 2^2 + 3^2.
%e A274274 a(31) = 1 since 31 = 3^3 + 0^2 + 2^2.
%t A274274 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A274274 Do[r=0;Do[If[SQ[n-x^3-y^2],r=r+1],{x,0,n^(1/3)},{y,0,Sqrt[(n-x^3)/2]}];Print[n," ",r];Continue,{n,0,80}]
%Y A274274 Cf. A000290, A000578, A022551, A022552, A262857, A272979.
%K A274274 nonn
%O A274274 0,2
%A A274274 _Zhi-Wei Sun_, Jul 14 2016