A300666 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w and z <= w such that x or 2*y is a square and x + 3*y is also a square.

1, 2, 3, 2, 2, 2, 2, 1, 1, 3, 4, 3, 1, 3, 3, 2, 2, 5, 6, 3, 3, 3, 3, 1, 1, 4, 7, 5, 2, 4, 3, 3, 3, 4, 7, 3, 5, 3, 5, 3, 2, 6, 6, 3, 1, 5, 3, 2, 2, 4, 7, 5, 6, 4, 4, 2, 1, 3, 5, 3, 2, 4, 4, 4, 2, 7, 10, 6, 3, 5, 4, 2, 2, 6, 9, 5, 3, 4, 5, 1, 2

Offset: 0

Author

Zhi-Wei Sun, Mar 10 2018

Keywords

• nonn

Comments

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

Conjecture 2: Any nonnegative integer n not equal to 3 can be written as x^2 + y^2 + z^2 + w^2 with nonnegative integers x,y,z,w such that x or 2*y is a square and 3*x - y is also a square.

By the author's 2017 JNT paper, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with integers x,y,z,w such that x (or 2*x) is a square.

See also A281976, A300667, A300708 and A300712 for similar conjectures.

a(n) > 0 for all n = 0..10^8. - Zhi-Wei Sun, Oct 04 2020

Examples

			a(8) = 1 since 8 = 0^2 + 0^2 + 2^2 + 2^2 with 0 = 0^2 and 0 + 3*0 = 0^2.
a(23) = 1 since 23 = 3^2 + 2^2 + 1^2 + 3^2 with 2*2 = 2^2 and 3 + 3*2 = 3^2.
a(56) = 1 since 56 = 4^2 + 0^2 + 2^2 + 6^2 with 4 = 2^2 and 4 + 3*0 = 2^2.
a(140) = 1 since 140 = 10^2 + 2^2 + 0^2 + 6^2 with 2*2 = 2^2 and 10 + 3*2 = 4^2.
a(472) = 1 since 472 = 0^2 + 12^2 + 2^2 + 18^2 with 0 = 0^2 and 0 + 3*12 = 6^2.
a(959) = 1 since 959 = 9^2 + 9^2 + 11^2 + 26^2 with 9 = 3^2 and 9 + 3*9 = 6^2.
a(1839) = 1 since 1839 = 1^2 + 5^2 + 7^2 + 42^2 with 1 = 1^2 and 1 + 3*5 = 4^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Yue-Feng She and Hai-Liang Wu, Proof of a conjecture of Sun on sums of four squares, arXiv:2010.02067 [math.NT], 2020.
• Yue-Feng She and Hai-Liang Wu, Sums of four squares with a certain restriction, Bull. of the Australian Math. Soc. (2021) First View, 1-10.
• 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], 2017-2018.

Crossrefs

Cf. A000118, A000290, A271518, A281976, A300667, A300708, A300712.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[(SQ[m^2-3y]||SQ[2y])&&SQ[n-(m^2-3y)^2-y^2-z^2],r=r+1],{m,0,(10n)^(1/4)},{y,0,Min[m^2/3,Sqrt[n]]},{z,0,Sqrt[Max[0,(n-(m^2-3y)^2-y^2)/2]]}];tab=Append[tab,r],{n,0,80}];Print[tab]