A283205 Numbers of the form x^2 + y^2 with x and y integers such that x + 3*y is a square.

0, 1, 2, 5, 8, 9, 10, 13, 16, 17, 25, 26, 29, 32, 34, 37, 40, 50, 53, 58, 61, 65, 73, 74, 80, 81, 85, 90, 104, 109, 117, 125, 128, 130, 136, 137, 144, 145, 146, 160, 162, 170, 178, 185, 193, 202, 208, 221, 229, 232, 241, 245, 250, 256, 257, 265, 269, 272, 274, 281

Offset: 1

Author

Zhi-Wei Sun, Mar 03 2017

Keywords

• nonn

Comments

This sequence is interesting since part (ii) of the conjecture in A283170 implies that each n = 0,1,2,... can be expressed as the sum of two terms of the current sequence.

Clearly, the sequence is a subsequence of A001481.

Examples

			a(1) = 0 since 0 = 0^2 + 0^2 with 0 + 3*0 = 0^2.
a(2) = 1 since 1 = 1^2 + 0^2 with 1 + 3*0 = 1^2.
a(3) = 2 since 2 = 1^2 + 1^2 with 1 + 3*1 = 2^2.
a(4) = 5 since 5 = (-2)^2 + 1^2 with (-2) + 3*1 = 1^2.
a(5) = 8 since 8 = (-2)^2 + 2^2 with (-2) + 3*2 = 2^2.
a(6) = 9 since 9 = 0^2 + 3^2 with 0 + 3*3 = 3^2.
a(7) = 10 since 10 = 3^2 + (-1)^2 with 3 + 3*(-1) = 0^2.
		

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. A000290, A001481, A281939, A283170, A283204.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
n=0;Do[Do[If[SQ[m-x^2],Do[If[SQ[(-1)^i*x+3(-1)^j*Sqrt[m-x^2]],n=n+1;Print[n," ",m];Goto[aa]],{i,0,Min[x,1]},{j,0,Min[Sqrt[m-x^2],1]}]],{x,0,Sqrt[m]}];Label[aa];Continue,{m,0,281}]