%I #21 Jan 19 2021 21:01:31
%S 1,2,4,1,2,3,4,2,5,3,4,2,3,3,4,1,2,3,4,2,6,3,3,3,4,5,6,4,6,6,5,3,9,5,
%T 4,2,4,5,6,2,5,4,5,3,6,4,4,5,5,3,6,5,4,3,4,2,6,5,4,2,3,3,7,5,4,6,5,4,
%U 7,1,2,3,6,4,3,3,5,5,4,2,6,2,5,3,2,8,7,5,6,6,6,4,10,8,7,4,4,9,8,6,10
%N Number of ways to write 2*n + 1 as x^2 + y^2 + z^2 + w^2 with x + y + 2*z a positive power of two, where x, y, z, w are nonnegative integers with x <= y.
%C Conjecture: a(n) > 0 for all n >= 0. Moreover, any integer m > 10840 not congruent to 0 or 2 modulo 8 can be written as x^2 + y^2 + z^2 + w^2 with x + y + 2*z = 4^k for some positive integer k, where x, y, z, w are nonnegative integers.
%C We have verified the latter assertion in the conjecture for m up to 5*10^6. By Theorem 1.4(i) of the author's 2019 IJNT paper, any positive integer m can be written as x^2 + y^2 + z^2 + w^2 with x, y, z, w integers such that x + y + 2*z = 4^k for some nonnegative integer k.
%C See also A338094 and A338096 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A338095/b338095.txt">Table of n, a(n) for n = 0..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. See also <a href="http://arxiv.org/abs/1604.06723">arXiv:1604.06723 [math.NT]</a>.
%H Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893. See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.
%H Zhi-Wei Sun, <a href="https://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.
%e a(3) = 1, and 2*3 + 1 = 1^2 + 1^2 + 1^2 + 2^2 with 1 + 1 + 2*1 = 2^2.
%e a(15) = 1, and 2*15 + 1 = 1^2 + 5^2 + 1^2 + 2^2 with 1 + 5 + 2*1 = 2^3.
%e a(69) = 1, and 2*69 + 1 = 7^2 + 9^2 + 0^2 + 3^2 with 7 + 9 + 2*0 = 2^4.
%e a(315) = 1, and 2*315 + 1 = 3^2 + 9^2 + 10^2 + 21^2 with 3 + 9 + 2*10 = 2^5.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t PQ[n_]:=PQ[n]=n>1&&IntegerQ[Log[2,n]];
%t tab={};Do[r=0;Do[If[SQ[2n+1-x^2-y^2-z^2]&&PQ[x+y+2z],r=r+1],{x,0,Sqrt[(2n+1)/2]},{y,x,Sqrt[2n+1-x^2]},{z,Boole[x+y==0],Sqrt[2n+1-x^2-y^2]}];
%t tab=Append[tab,r],{n,0,100}];Print[tab]
%Y Cf. A000079, A000118, A000290, A000302, A279612, A338094, A338096, A338119, A338121.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Oct 09 2020