A283204 - OEIS

%I #12 Mar 03 2017 02:49:26

%S 0,1,2,4,5,10,13,16,17,18,20,26,29,32,34,37,45,50,52,53,58,61,64,65,

%T 68,74,80,81,85,97,100,106,109,113,116,122,125,130,145,146,148,149,

%U 157,160,162,170,173,180,197,205,208,218,221,234,245,250,256,260,261,269

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

%C 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.

%C Clearly, the sequence is a subsequence of A001481. See also A283205 for a similar sequence.

%H Zhi-Wei Sun, <a href="/A283204/b283204.txt">Table of n, a(n) for n = 1..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.

%e a(1) = 0 since 0 = 0^2 + 0^2 with 0 + 2*0 = 0^2.

%e a(2) = 1 since 1 = 1^2 + 0^2 with 1 + 2*0 = 1^2.

%e a(3) = 2 since 2 = (-1)^2 + 1^2 with (-1) + 2*1 = 1^2.

%e a(4) = 4 since 4 = 0^2 + 2^2 with 0 + 2*2 = 2^2.

%e a(5) = 5 since 5 = 2^2 + 1^2 with 2 + 2*1 = 2^2.

%e a(6) = 10 since 10 = 3^2 + (-1)^2 with 3 + 2*(-1) = 1^2.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t 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}]

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

%K nonn

%O 1,3

%A _Zhi-Wei Sun_, Mar 03 2017