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%I #13 Nov 23 2016 14:22:52
%S 0,1,2,3,8,9,10,13,14,16,17,19,21,25,26,29,30,32,37,38,40,41,42,46,48,
%T 49,50,51,54,58,59,65,66,69,70,72,73,74,77,78,81,83,85,89,90,97,98,
%U 101,102,104,105,106,109,114,117,118,120,122,125,128,129,130,131,134,136,138,139,144,145,146
%N Numbers of the form x^2 + y^2 + z^2 with x + 3*y + 5*z a square, where x, y and z are nonnegative integers.
%C This is motivated by the author's 1-3-5-Conjecture which states that any nonnegative integer can be expressed as the sum of a square and a term of the current sequence.
%C Clearly, any term times a fourth power is also a term of this sequence. By the Gauss-Legendre theorem on sums of three squares, no term has the form 4^k*(8m+7) with k and m nonnegative integers.
%H Zhi-Wei Sun, <a href="/A278560/b278560.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e a(4) = 3 since 3 = 1^2 + 1^2 + 1^2 with 1 + 3*1 + 5*1 = 3^2.
%e a(5) = 8 since 8 = 0^2 + 2^2 + 2^2 with 0 + 3*2 + 5*2 = 4^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t n=0;Do[Do[If[SQ[m-x^2-y^2]&&SQ[x+3y+5*Sqrt[m-x^2-y^2]],n=n+1;Print[n," ",m];Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]}];Label[aa];Continue,{m,0,146}]
%Y Cf. A000290, A271518, A273294, A273302.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Nov 23 2016
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