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%I #11 Oct 04 2020 23:43:36
%S 1,1,2,1,3,2,2,1,2,2,3,2,3,2,2,2,1,2,4,3,5,1,3,2,1,2,3,4,3,3,2,2,2,2,
%T 4,3,6,2,3,2,3,3,3,3,3,3,1,3,1,3,5,2,6,3,5,3,2,4,3,2,4,3,3,3,3,5,3,3,
%U 8,5,3,2,5,3,4,3,3,5,3,1,2
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 3*x or y is a square and x - y is twice a square.
%C Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 16^k*m with k = 0,1,2,... and m = 0, 1, 3, 7, 21, 24, 46, 79, 88, 94, 142, 151, 184, 190, 193, 280, 286, 1336.
%C By the author's 2017 JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 2*(x-y) (or x) is a square.
%C See also A281976, A300666, A300667 and A300708 for similar conjectures.
%C a(n) > 0 for all n = 0..10^8. - _Zhi-Wei Sun_, Oct 04 2020
%H Zhi-Wei Sun, <a href="/A300712/b300712.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.
%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(21) = 1 since 21 = 2^2 + 0^2 + 1^2 + 4^2 with 0 = 0^2 and 2 - 0 = 2*1^2.
%e a(79) = 1 since 79 = 3^2 + 3^2 + 5^2 + 6^2 with 3*3 = 3^2 and 3 - 3 = 2*0^2.
%e a(142) = 1 since 142 = 6^2 + 4^2 + 3^2 + 9^2 with 4 = 2^2 and 6 - 4 = 2*1^2.
%e a(190) = 1 since 190 = 3^2 + 1^2 + 6^2 + 12^2 with 1 = 1^2 and 3 - 1 = 2*1^2.
%e a(193) = 1 since 193 = 0^2 + 0^2 + 7^2 + 12^2 with 0 = 0^2 and 0 - 0 = 2*0^2.
%e a(280) = 1 since 280 = 12^2 + 10^2 + 0^2 + 6^2 with 3*12 = 6^2 and 12 - 10 = 2*1^2.
%e a(1336) = 1 since 1336 = 2^2 + 0^2 + 6^2 + 36^2 with 0 = 0^2 and 2 - 0 = 2*1^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[(SQ[3(2m^2+y)]||SQ[y])&&SQ[n-(2m^2+y)^2-y^2-z^2],r=r+1],{m,0,(n/4)^(1/4)},{y,0,Sqrt[(n-4m^4)/2]},{z,0,Sqrt[Max[0,(n-(2m^2+y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]
%Y Cf. A000118, A000290, A271518, A271775, A281976, A300666, A300667, A300708.
%K nonn
%O 0,3
%A _Zhi-Wei Sun_, Mar 11 2018
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