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%I #41 Oct 29 2020 20:37:14
%S 1,1,2,3,1,2,1,1,1,1,4,1,2,2,2,1,6,3,7,3,2,7,3,6,4,5,3,3,1,1,3,6,4,5,
%T 9,1,6,4,7,2,4,4,6,3,1,6,3,5,5,4,1,3,4,4,6,7,4,3,5,3,9,3,6,3,1,10,7,2,
%U 8,3,2,10,6,5,3,4,5,4,5,5,4
%N Number of ways to write 8*n+7 as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(3*x+7*y+10*z) is a square and also one of w, x, y, z is a square.
%C Conjecture 1: Each nonnegative integer can be written as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(3*x+7*y+10*z) is a square and one of w, x, y, z is also a square.
%C As all the nonnegative integers not of the form 4^k*(8*m+7) (k>=0, m>=0) can be written as 2*0^2 + x^2 + y^2 + z^2 with x, y, z integers, Conjecture 1 has the following equivalent version: a(n) > 0 for all n = 0,1,...
%C We have verified that a(n) > 0 for all n = 0..10^5.
%C Conjecture 2: If (a,b) is among the ordered pairs (1,2), (1,3), (2,4), (2,5), (2,8), (2,24), (6,8), (6,32), (9,12) and (18,24), then each n = 0,1,... can be written as 2*w^2 + x^2 + y^2 + z^2 with w, x, y, z nonnegative integers such that w*(a*x+b*y) is a square.
%H Zhi-Wei Sun, <a href="/A338263/b338263.txt">Table of n, a(n) for n = 0..6000</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="http://arxiv.org/abs/2010.05775">Sums of four squares with certain restrictions</a>, arXiv:2010.05775 [math.NT], 2020.
%e a(7) = 1, and 8*7+7 = 63 = 2*3^2 + 6^2 + 0^2 + 3^2 with 0 = 0^2 and 3*(3*6+7*0+10*3) = 12^2.
%e a(11) = 1, and 8*11+7 = 95 = 2*1^2 + 2^2 + 5^2 + 8^2 with 1 = 1^2 and 1*(3*2+7*5+10*8) = 11^2.
%e a(15) = 1, and 8*15+7 = 127 = 2*7^2 + 3^2 + 2^2 + 4^2 with 4 = 2^2 and 7*(3*3+7*2+10*4) = 21^2.
%e a(64) = 1, and 8*64+7 = 519 = 2*3^2 + 1^2 + 20^2 + 10^2 with 1 = 1^2 and 3*(3*1+7*20+10*10) = 27^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[8n+7-2w^2-x^2-y^2]&&(SQ[w]||SQ[x]||SQ[y]||SQ[8n+7-2w^2-x^2-y^2])&&SQ[w(3x+7y+10*Sqrt[8n+7-2w^2-x^2-y^2])],r=r+1],{w,0,Sqrt[4n+3]},{x,0,Sqrt[8n+7-2w^2]},{y,0,Sqrt[8n+7-2w^2-x^2]}];tab=Append[tab,r],{n,0,80}];tab
%Y Cf. A000290, A271518, A271724, A275301, A275344.
%K nonn
%O 0,3
%A _Zhi-Wei Sun_, Oct 27 2020
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