cp's OEIS Frontend

A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y and 2*x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer.

%I A283196 #17 Mar 02 2017 23:05:07
%S A283196 1,1,1,2,2,1,3,1,1,8,1,1,6,1,3,1,3,9,2,3,3,4,4,1,7,5,2,4,3,3,6,1,5,7,
%T A283196 1,5,4,6,4,3,2,8,3,2,11,2,6,1,6,5,1,9,4,7,11,1,3,16,1,2,5,3,14,2,7,7,
%U A283196 4,6,3,12,6,3,8,5,2,3,5,5,9,2
%N A283196 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 2*x + y and 2*x + z both squares, where x,y,z are integers with |y| <= |z|, and w is a positive integer.
%C A283196 Conjecture: (i) a(n) > 0 for all n > 0.
%C A283196 (ii) Any positive integer n can be written as x^2 + y^2 + z^2 + w^2 such that both x + y and x + 2*z are squares, where x,y,z,w are integers with x >= 0 and w > 0.
%C A283196 (iii)  Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with 2*x + 2*y and 2*x + z both squares, where x,y,z,w are integers with x*y <= 0.
%C A283196 (iv) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with 2*|x-y| and 2*x + z both squares, where x,y,z,w are integers with x >= 0 and y >= 0.
%C A283196 By the linked JNT paper, any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that 2*x + y is a square, and also we can write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x - y (or 2*x - 2*y) is a square.
%H A283196 Zhi-Wei Sun, <a href="/A283196/b283196.txt">Table of n, a(n) for n = 1..10000</a>
%H A283196 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 A283196 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e A283196 a(2) = 1 since 2 = 0^2 + 0^2 + 1^2 + 1^2 with 2*0 + 0 = 0^2 and 2*0 + 1 = 1^2.
%e A283196 a(14) = 1 since 14 = 2^2 + 0^2 + (-3)^2 + 1^2 with 2*2 + 0 = 2^2 and 2*2 + (-3) = 1^2.
%e A283196 a(59) = 1 since 59 = 3^2 + 3^2 + (-5)^2 + 4^2 with 2*3 + 3 = 3^2 and 2*3 + (-5) = 1^2.
%e A283196 a(88) = 1 since 88 = (-2)^2 + 4^2 + 8^2 + 2^2 with 2*(-2) + 4 = 0^2 and 2*(-2) + 8 = 2^2.
%e A283196 a(131) = 1 since 131 = 0^2 + 1^2 + 9^2 + 7^2 with 2*0 + 1 = 1^2 and 2*0 + 9 = 3^2.
%e A283196 a(219) = 1 since 219 = 8^2 + (-7)^2 + 9^2 + 5^2 with 2*8 + (-7) = 3^2 and 2*8 + 9 = 5^2.
%e A283196 a(249) = 1 since 249 = (-4)^2 + 8^2 + 12^2 + 5^2 with 2*(-4) + 8 = 0^2 and 2*(-4) + 12 = 2^2.
%e A283196 a(312) = 1 since 312 = 6^2 + 4^2 + (-8)^2 + 14^2 with 2*6 + 4 = 4^2 and 2*6 + (-8) = 2^2.
%e A283196 a(323) = 1 since 323 = 9^2 + 7^2 + 7^2 + 12^2 with 2*9 + 7 = 5^2.
%e A283196 a(536) = 1 since 536 = (-6)^2 + 12^2 + 16^2 + 10^2 with 2*(-6) + 12 = 0^2 and 2*(-6) + 16 = 2^2.
%e A283196 a(888) = 1 since 888 = 14^2 + 8^2 + (-12)^2 + 22^2 with 2*14 + 8 = 6^2 and 2*14 + (-12) = 4^2.
%e A283196 a(1464) = 1 since 1464 = 2^2 + 0^2 + (-4)^2 + 38^2 with 2*2 + 0 = 2^2 and 2*2 + (-4) = 0^2.
%e A283196 a(4152) = 1 since 4152 = 30^2 + 4^2 + (-56)^2 + 10^2 with 2*30 + 4 = 8^2 and 2*30 + (-56) = 2^2.
%t A283196 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A283196 Do[r=0;Do[If[SQ[2(-1)^i*x+(-1)^j*y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[2(-1)^i*x+(-1)^k*z],r=r+1],{z,y,Sqrt[n-1-x^2-y^2]},{k,0,Min[z,1]}]],{x,0,Sqrt[n-1]},{y,0,Sqrt[(n-1-x^2)/2]},{i,0,Min[x,1]},{j,0,Min[y,1]}];Print[n," ",r];Continue,{n,1,80}]
%Y A283196 Cf. A000118, A000290, A271518, A281975, A281976, A281939, A283170.
%K A283196 nonn
%O A283196 1,4
%A A283196 _Zhi-Wei Sun_, Mar 02 2017