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 A273404 #6 May 21 2016 23:26:47
%S A273404 1,2,3,2,2,3,3,2,1,3,4,2,1,2,2,2,2,3,5,2,3,3,2,1,1,4,5,4,2,2,4,3,3,3,
%T A273404 6,2,6,5,3,3,3,7,6,2,2,5,4,1,2,3,7,6,8,4,5,5,2,4,5,2,3,5,3,4,2,5,9,4,
%U A273404 5,4,5,1,3,5,4,5,5,4,2,3,3
%N A273404 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x + 24*y a square, where x,y,z,w are nonnegative integers with z <= w.
%C A273404 Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 12, 23, 24, 47, 71, 168, 344, 632).
%C A273404 For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.
%H A273404 Zhi-Wei Sun, <a href="/A273404/b273404.txt">Table of n, a(n) for n = 0..10000</a>
%H A273404 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e A273404 a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 + 24*0 = 0^2.
%e A273404 a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 24*1 = 5^2.
%e A273404 a(23) = 1 since 23 = 1^2 + 2^2 + 3^2 + 3^2 with 1 + 24*2 = 7^2.
%e A273404 a(24) = 1 since 24 = 4^2 + 0^2 + 2^2 + 2^2 with 4 + 24*0 = 2^2.
%e A273404 a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 + 24*1 = 5^2.
%e A273404 a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 + 24*5 = 11^2.
%e A273404 a(168) = 1 since 168 = 4^2 + 4^2 + 6^2 + 10^2 with 4 + 24*4 = 10^2.
%e A273404 a(344) = 1 since 344 = 4^2 + 0^2 + 2^2 + 18^2 with 4 + 24*0 = 2^2.
%e A273404 a(632) = 1 since 632 = 0^2 + 6^2 + 14^2 + 20^2 with 0 + 24*6 = 12^2.
%t A273404 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A273404 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x+24y],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[(n-x^2-y^2)/2]}];Print[n," ",r];Label[aa];Continue,{n,0,80}]
%Y A273404 Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302.
%K A273404 nonn
%O A273404 0,2
%A A273404 _Zhi-Wei Sun_, May 21 2016