A283170 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x + 2*y and z + 2*w both squares, where x and w are nonnegative integers, and y and z are integers.

1, 2, 2, 1, 2, 6, 5, 2, 1, 4, 5, 1, 1, 2, 3, 2, 2, 6, 9, 3, 5, 8, 7, 6, 2, 4, 2, 2, 1, 5, 7, 6, 2, 8, 10, 3, 5, 5, 4, 4, 1, 1, 8, 1, 2, 6, 7, 4, 1, 7, 12, 3, 5, 5, 7, 10, 2, 7, 9, 1, 2, 8, 8, 12, 2, 11, 13, 2, 7, 11, 9, 8, 3, 4, 10, 3, 2, 6, 6, 10, 6

Offset: 0

Author

Zhi-Wei Sun, Mar 02 2017

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n = 0,1,2,....

(ii) Any nonnegative integer n can be written as x^2 + y^2 + z^2 + w^2 with x + 3*y and z + 3*w both squares, where x,y,z are integers and w is a nonnegative integer.

(iii) Every nonnegative integer can be expressed as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z + 3*w are squares.

(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is a square and |2*z-w| is twice a square. Also, each nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z nonnegative integers and w an integer such that |2*x-y| is twice a square and |2*z-w| is a square.

(v) Every 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-2*y)*(z-2*w)| is twice a square. Also, any positive integer n can be written as x^2 + y^2 + z^2 + w^2 with x a positive integer and y,z,w nonnegative integers such that (2*x+y)*(2*z-w) is twice a square.

(vi) Each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that both x + 2*y and z^2 - w^2 (or z^2 + 8*w^2, or 7*z^2 + 9*w^2) are squares.

(vii) Any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both 2*x - y and 64*z^2 - 84*z*w + 21*w^2 (or 81*z^2 - 112*z*w + 56*w^2) are squares.

By the linked JNT paper, any nonnegative integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + 2*y is a square.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Crossrefs

Cf. A000118, A000290, A271518, A281976, A281939, A282561, A282562.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[x+2(-1)^j*y],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(-1)^k*z+2*Sqrt[n-x^2-y^2-z^2]],r=r+1],{z,0,Sqrt[n-x^2-y^2]},{k,0,Min[z,1]}]],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{j,0,Min[y,1]}];Print[n," ",r];Continue,{n,0,80}]

Formula

a(3) = 1 since 3 = 1^2 + 0^2 + (-1)^2 + 1^2 with 1 + 2*0 = 1^2 and (-1)+2*1 = 1^2.

a(4) = 2 since 4 = 0^2 + 0^2 + 0^2 + 2^2 with 0 + 2*0 = 0^2 and 0 + 2*2 = 2^2, and also 4 = 0^2 + 2^2 + 0^2 + 0^2 with 0 + 2*2 = 2^2 and 0 + 2*0 = 0^2.

a(8) = 1 since 8 = 0^2 + 2^2 + 0^2 + 2^2 with 0 + 2*2 = 2^2 and 0 + 2*2 = 2^2.

a(11) = 1 since 11 = 3^2 + (-1)^2 + 1^2 + 0^2 with 3 + 2*(-1) = 1^2 and 1 + 2*0 = 1^2.

a(12) = 1 since 12 = 3^2 + (-1)^2 + (-1)^2 + 1^2 with 3 + 2*(-1) = 1^2 and (-1) + 2*1 = 1^2.

a(28) = 1 since 28 = 3^2 + (-1)^2 + 3^2 + 3^2 with 3 + 2*(-1) = 1^2 and 3 + 2*3 = 3^2.

a(40) = 1 since 40 = 4^2 + (-2)^2 + (-4)^2 + 2^2 with 4 + 2*(-2) = 0^2 and (-4) + 2*2 = 0^2.

a(41) = 1 since 41 = 6^2 + (-1)^2 + 0^2 + 2^2 with 6 + 2*(-1) = 2^2 and 0 + 2*2 = 2^2.

a(332) = 1 since 332 = 11^2 + 7^2 + (-9)^2 + 9^2 with 11 + 2*7 = 5^2 and (-9) + 2*9 = 3^2.

a(443) = 1 since 443 = 19^2 + (-9)^2 + 1^2 + 0^2 with 19 + 2*(-9) = 1^2 and 1 + 2*0 = 1^2.

a(488) = 1 since 488 = 12^2 + 2^2 + (-12)^2 + 14^2 with 12 + 2*2 = 4^2 and (-12) + 2*14 = 4^2.

a(808) = 1 since 808 = 8^2 + 14^2 + (-8)^2 + 22^2 with 8 + 2*14 = 6^2 and (-8) + 2*22 = 6^2.

a(892) = 1 since 892 = 27^2 + (-1)^2 + (-9)^2 + 9^2 with 27 + 2*(-1) = 5^2 and (-9) + 2*9 = 3^2.