%I #16 Mar 20 2018 16:11:10

%S 1,1,2,1,1,3,1,1,4,2,2,3,3,3,3,1,5,6,2,2,10,5,4,3,2,7,7,3,5,4,3,1,12,

%T 8,2,6,4,5,10,2,7,13,8,5,10,6,6,3,8,4,7,7,8,11,4,3,17,9,5,4,8,5,9,1,8,

%U 14,8,8,13,5,8,6,11,10,7,5,13,15,7,2

%N Number of ways to write n^2 as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that x^2-(3*y)^2 = 4^k for some k = 0,1,2,....

%C Conjecture: a(n) > 0 for all n > 0. Moreover, any positive square n^2 can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers and y even such that x^2 - (3*y)^2 = 4^k for some k = 0,1,2,....

%C We have verifed this for all n = 1..10^7.

%C Compare this conjecture with the conjectures in A299537.

%C As 3*A001353(n)^2 + 1 = A001075(n)^2, the conjecture in A300441 implies that any positive square can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x^2 - 3*y^2 = 4^k for some k = 0,1,2,....

%C See also A301391 for a similar conjecture.

%H Zhi-Wei Sun, <a href="/A301376/b301376.txt">Table of n, a(n) for n = 1..10000</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], 2017-2018.

%e a(1) = 1 since 1^2 = 1^2 + 0^2 + 0^2 + 0^2 with 1^2 - (3*0)^2 = 4^0.

%e a(5) = 1 since 5^2 = 4^2 + 0^2 + 0^2 + 3^2 with 4^2 - (3*0)^2 = 4^2.

%e a(7) = 1 since 7^2 = 2^2 + 0^2 + 3^2 + 6^2 with 2^2 - (3*0)^2 = 4^1.

%e a(31) = 3 since 31^2 = 10^2 + 2^2 + 4^2 + 29^2 with 10^2 - (3*2)^2 = 4^3, and 31^2 = 20^2 + 4^2 + 4^2 + 23^2 = 20^2 + 4^2 + 16^2 + 17^2 with 20^2 - (3*4)^2 = 4^4.

%t f[n_]:=f[n]=FactorInteger[n];

%t g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;

%t QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t tab={};Do[r=0;Do[If[SQ[4^k+9y^2]&&QQ[n^2-4^k-10y^2],Do[If[SQ[n^2-(4^k+10y^2)-z^2],r=r+1],{z,0,Sqrt[(n^2-4^k-10y^2)/2]}]],{k,0,Log[2,n]},{y,0,Sqrt[(n^2-4^k)/10]}];tab=Append[tab,r],{n,1,80}];Print[tab]

%Y Cf. A000118, A000290, A000302, A299537, A299794, A299924, A300219, A300396, A300441, A300510, A301391.

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 19 2018