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A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 16*y^2 + 24*z^2 + 48*w^2 a square, where x,y,z,w are nonnegative integers.

%I A281659 #17 Feb 15 2017 00:34:15
%S A281659 1,2,2,2,2,1,4,1,2,3,4,1,5,1,5,1,2,2,8,1,2,5,2,2,4,3,2,8,1,5,4,2,2,3,
%T A281659 3,4,4,3,2,2,4,2,6,1,1,3,2,3,5,2,3,7,3,1,7,1,5,2,2,4,1,1,6,7,2,1,7,2,
%U A281659 4,5,6,0,8,3,6,8,5,3,4,2,2
%N A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 16*y^2 + 24*z^2 + 48*w^2 a square, where x,y,z,w are nonnegative integers.
%C A281659 Conjecture: (i) a(n) > 0 except for n = 71, 85.
%C A281659 (ii) Any nonnegative integer n not among 39, 71 and 649 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 16*y^2 + 24*z^2 + 32*w^2 a square, where x,y,z,w are integers.
%C A281659 (iii) Each nonnegative integer n not among 5, 7, 23, 47, 93, 103, 109, 151, 191 and 911 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 3*y^2 + 8*z^2 + 16*w^2 a square, where x,y,z,w are integers.
%C A281659 (iv) Any nonnegative integer n not among 15, 19, 71, 103, 191 and 559 can be written as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 48*y^2 + 64*z^2 + 96*w^2 a square, where x,y,z,w are integers.
%C A281659 (v) Any nonnegative integer n not among 5, 93, 95, 161 and 309 can be written as x^2 + y^2 + z^2 + w^2 with 16*x^2 + 25*y^2 + 48*z^2 + 128*w^2 a square, where x,y,z,w are integers.
%C A281659 (vi) Each nonnegative integer not among 13, 19, 39, 41, 71, 109, 131, 193, 233, 377, 415 and 941 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 24*y^2 + 48*z^2 + 144*w^2 a square, where x,y,z,w are integers.
%C A281659 We have verified part (i) of the conjecture for n up to 1.2*10^5.
%H A281659 Zhi-Wei Sun, <a href="/A281659/b281659.txt">Table of n, a(n) for n = 0..10000</a>
%H A281659 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 A281659 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e A281659 a(11) = 1 since 11 = 3^2 + 1^2 + 1^2 + 0^2 with 9*3^2 + 16*1^2 + 24*1^2 + 48*0 = 11^2.
%e A281659 a(170) = 1 since 170 = 3^2 + 6^2 + 2^2 + 11^2 with 9*3^2 + 16*6^2 + 24*2^2 + 48*11^2 = 81^2.
%e A281659 a(305) = 1 since 305 = 0^2 + 15^2 + 4^2 + 8^2 with 9*0^2 + 16*15^2 + 24*4^2 + 48*8^2 = 84^2.
%t A281659 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A281659 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9x^2+16y^2+24z^2+48*(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Label[aa];Continue,{n,0,80}]
%Y A281659 Cf. A000118, A000290, A271510, A271513, A271518.
%K A281659 nonn
%O A281659 0,2
%A A281659 _Zhi-Wei Sun_, Jan 26 2017