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%I #11 Feb 10 2017 07:43:42
%S 0,1,2,10,18,34,41,52,66,100,90,130,261,306,226,370,426,405,612,585,
%T 661,626,770,666,756,706,810,981,882,1026,1266,1170,1330,1530,1476,
%U 1426,1881,1701,2650,2410,2506,1666,1386,2226,3861,2626,3366,3006,2106,2610,3346,3186,3226,4410,3786,3850,2826,3762,4026,4500
%N Least nonnegative integer m which can be written in exactly n ways as x^2 + y^2 + z^2 + w^2 with both x and x + 24*y squares, where x,y,z,w are nonnegative integers with z <= w.
%C Conjecture: a(n) exists for any n > 0.
%C See also A281976 for a related conjecture.
%H Zhi-Wei Sun, <a href="/A282226/b282226.txt">Table of n, a(n) for n = 1..300</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e a(1) = 0 since 0 = 0^2 + 0^2 + 0^2 + 0^2 with 0 = 0^2 and 0 + 24*0 = 0^2.
%e a(2) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 0 = 0^2 and 0 + 24*0 = 0^2, and 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 = 1^2 and 1 + 24*0 = 1^2.
%e a(4) = 10 since 10 = 0^2 + 0^2 + 1^2 + 3^2 with 0 = 0^2 and 0 + 24*0 = 0^2, 10 = 1^2 + 0^2 + 0^2 + 3^2 with 1 = 1^2 and 1 + 24*0 = 1^2, 10 = 1^2 + 1^2 + 2^2 + 2^2 with 1 = 1^2 and 1 + 24*1 = 5^2, and 10 = 1^2 + 2^2 + 1^2 + 2^2 with 1 = 1^2 and 1 + 24*2 = 7^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[m=0;Label[aa];r=0;Do[If[SQ[m-x^4-y^2-z^2]&&SQ[x^2+24y],r=r+1;If[r>n,m=m+1;Goto[aa]]],{x,0,m^(1/4)},{y,0,Sqrt[m-x^4]},{z,0,Sqrt[(m-x^4-y^2)/2]}];If[r<n,m=m+1;Goto[aa],Print[n," ",m]];Continue,{n,1,60}]
%Y Cf. A000118, A000290, A270969, A281976, A281980.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Feb 09 2017
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