A281729 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,z positive integers and y,w nonnegative integers such that both 9*x^2 + 246*x*y + y^2 and 9*z^2 + 666*z*w + w^2 are squares.

0, 1, 2, 1, 2, 4, 2, 1, 4, 5, 2, 2, 4, 3, 1, 1, 5, 5, 3, 4, 6, 6, 2, 4, 5, 5, 5, 3, 5, 5, 3, 1, 7, 8, 2, 5, 6, 7, 2, 5, 7, 4, 2, 3, 7, 4, 3, 2, 5, 7, 5, 6, 7, 12, 4, 3, 7, 7, 2, 1, 7, 5, 4, 1, 7, 7, 3, 7, 8, 6, 2, 5, 7, 6, 4, 4, 8, 4, 1, 4

Offset: 1

Author

Zhi-Wei Sun, Feb 19 2017

Keywords

• nonn

Comments

The first three values of n with a(n) = 0 are 1, 214635, 241483.

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 nonnegative integers and x*(x-y) = 0, Whether x = 0 or x = y, both 9*x^2 + 246*x*y + y^2 and 9*x^2 + 666*x*y + y^2 are squares.

Examples

			a(2) = 1 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 9*1^2 + 246*1*0 + 1^2 = 3^2 and 9*1^2 + 666*1*0 + 0^2 = 3^2.
a(4) = 1 since 4 = 1^2 + 1^2 + 1^2 + 1^2 with 9*1^2 + 246*1*1 + 1^2 = 16^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 9*2^2 + 246*2*3 + 3^2 = 39^2 and 9*1^2 + 666*1*1 + 1^2 = 26^2.
a(159) = 1 since 159 = 11^2 + 3^2 + 5^2 + 2^2 with 9*11^2 + 246*11*3 + 3^2 = 96^2 and 9*5^2 + 666*5*2 + 2^2 = 83^2.
a(515) = 1 since 515 = 15^2 + 0^2 + 17^2 + 1^2 with 9*15^2 + 246*15*0 + 0^2 = 45^2 and 9*17^2 + 666*17*1 + 1^2 = 118^2.
a(9795) = 1 since 9795 = 35^2 + 91^2 + 17^2 + 0^2 with 9*35^2 + 246*35*91 + 91^2 = 896^2 and 9*17^2 + 666*17*0 + 0^2 = 51^2.
a(84155) = 1 since 84155 = 281^2 + 0^2 + 35^2 + 63^2 with 9*281^2 + 246*281*0 + 0^2 = 843^2 and 9*35^2 + 666*35*63 + 63^2 = 1218^2.
a(121003) = 1 since 121003 = 319^2 + 87^2 + 3^2 + 108^2 with 9*319^2 + 246*319*87 + 87^2 = 2784^2 and 9*3^2 + 666*3*108 + 108^2 = 477^2.
a(133647) = 1 since 133647 = 217^2 + 217^2 + 115^2 + 162^2 with 9*217^2 + 246*217*217 + 217^2 = 3472^2 and 9*115^2 + 666*115*162 + 162^2 = 3543^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..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, A282561, A282562.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[9x^2+246x*y+y^2],Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9z^2+666z*Sqrt[n-x^2-y^2-z^2]+(n-x^2-y^2-z^2)],r=r+1],{z,1,Sqrt[n-x^2-y^2]}]],{x,1,Sqrt[n]},{y,0,Sqrt[n-x^2]}];Print[n," ",r];Continue,{n,1,80}]