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

1,3

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A225081 A307368 A097082 * A302290 A145173 A361656

Adjacent sequences: A281726 A281727 A281728 * A281730 A281731 A281732

Keyword

nonn

Author

Zhi-Wei Sun, Feb 19 2017

Status

approved