%I #11 Feb 04 2017 11:23:08
%S 1,3,4,3,2,4,4,2,1,4,5,3,1,2,3,2,3,5,7,3,4,5,2,1,2,6,9,6,2,4,6,3,4,6,
%T 8,4,6,5,4,3,2,11,10,4,1,7,5,1,3,3,11,9,7,5,6,4,2,7,5,4,2,6,4,3,2,7,
%U 15,4,4,6,5,1,2,6,7,7,3,6,4,2,4
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and z <= w such that both x and |x-y| are squares.
%C Conjecture: (i) a(n) > 0 for all n = 0,1,2,....
%C (ii) Each nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that |x-y| and 2*(y-z) (or 2*(z-y)) are both squares.
%C (iii) For each ordered pair (a,b) = (2,1), (3,1), (9,5), (14,10), any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x and |a*x-b*y| are both squares.
%C The author has proved that each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x (or x-y, or 2(x-y)) is a square.
%C See also A281976 and A281977 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A281975/b281975.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.
%e a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and |0-0| = 0^2.
%e a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 = 1^2 and |1-1| = 0^2.
%e a(44) = 1 since 44 = 1^2 + 5^2 + 3^2 + 3^2 with 1 = 1^2 and |1-5| = 2^2.
%e a(47) = 1 since 47 = 1^2 + 1^2 + 3^2 + 6^2 with 1 = 1^2 and |1-1| = 0^2.
%e a(71) = 1 since 71 = 1^2 + 5^2 + 3^2 + 6^2 with 1 = 1^2 and |1-5| = 2^2.
%e a(95) = 1 since 95 = 1^2 + 2^2 + 3^2 + 9^2 with 1 = 1^2 and |1-2| = 1^2.
%e a(140) = 1 since 140 = 9^2 + 5^2 + 3^2 + 5^2 with 9 = 3^2 and |9-5| = 2^2.
%e a(428) = 1 since 428 = 9^2 + 13^2 + 3^2 + 13^2 with 9 = 3^2 and |9-13| = 2^2.
%e a(568) = 1 since 568 = 4^2 + 8^2 + 2^2 + 22^2 with 4 = 2^2 and |4-8| = 2^2.
%e a(632) = 1 since 632 = 16^2 + 12^2 + 6^2 + 14^2 with 16 = 4^2 and |16-12| = 2^2.
%e a(1144) = 1 since 1144 = 16^2 + 20^2 + 2^2 + 22^2 with 16 = 4^2 and |16-20| = 2^2.
%e a(1544) = 1 since 1544 = 0^2 + 0^2 + 10^2 + 38^2 with 0 = 0^2 and |0-0| = 0^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[Abs[x^2-y]],r=r+1],{x,0,n^(1/4)},{y,0,Sqrt[n-x^4]},{z,0,Sqrt[(n-x^4-y^2)/2]}];Print[n," ",r];Continue,{n,0,80}]
%Y Cf. A000118, A000290, A270969, A271775, A281939, A281941, A281976, A281977.
%K nonn
%O 0,2
%A _Zhi-Wei Sun_, Feb 03 2017