A283204 Numbers of the form x^2 + y^2 with x and y integers such that x + 2*y is a square.

0, 1, 2, 4, 5, 10, 13, 16, 17, 18, 20, 26, 29, 32, 34, 37, 45, 50, 52, 53, 58, 61, 64, 65, 68, 74, 80, 81, 85, 97, 100, 106, 109, 113, 116, 122, 125, 130, 145, 146, 148, 149, 157, 160, 162, 170, 173, 180, 197, 205, 208, 218, 221, 234, 245, 250, 256, 260, 261, 269

Offset

1,3

Comments

This sequence is interesting since part (i) 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. See also A283205 for a similar sequence.

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

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
n=0; Do[Do[If[SQ[m-x^2], Do[If[SQ[(-1)^i*x+2(-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, 270}]

Crossrefs

Cf. A000290, A001481, A283170, A283196, A283205.

Sequence in context: A321683 A321682 A013578 * A105138 A326311 A325107

Adjacent sequences: A283201 A283202 A283203 * A283205 A283206 A283207

Keyword

nonn

Author

Zhi-Wei Sun, Mar 03 2017

Status

approved