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

%I A283205 #5 Mar 03 2017 02:48:35
%S A283205 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,
%T A283205 81,85,90,104,109,117,125,128,130,136,137,144,145,146,160,162,170,178,
%U A283205 185,193,202,208,221,229,232,241,245,250,256,257,265,269,272,274,281
%N A283205 Numbers of the form x^2 + y^2 with x and y integers such that x + 3*y is a square.
%C A283205 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.
%C A283205 Clearly, the sequence is a subsequence of A001481.
%H A283205 Zhi-Wei Sun, <a href="/A283205/b283205.txt">Table of n, a(n) for n = 1..10000</a>
%H A283205 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 A283205 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e A283205 a(1) = 0 since 0 = 0^2 + 0^2 with 0 + 3*0 = 0^2.
%e A283205 a(2) = 1 since 1 = 1^2 + 0^2 with 1 + 3*0 = 1^2.
%e A283205 a(3) = 2 since 2 = 1^2 + 1^2 with 1 + 3*1 = 2^2.
%e A283205 a(4) = 5 since 5 = (-2)^2 + 1^2 with (-2) + 3*1 = 1^2.
%e A283205 a(5) = 8 since 8 = (-2)^2 + 2^2 with (-2) + 3*2 = 2^2.
%e A283205 a(6) = 9 since 9 = 0^2 + 3^2 with 0 + 3*3 = 3^2.
%e A283205 a(7) = 10 since 10 = 3^2 + (-1)^2 with 3 + 3*(-1) = 0^2.
%t A283205 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A283205 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}]
%Y A283205 Cf. A000290, A001481, A281939, A283170, A283204.
%K A283205 nonn
%O A283205 1,3
%A A283205 _Zhi-Wei Sun_, Mar 03 2017