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Numbers of the form x^2 + y^2 + z^2 with x + 2*y - 2*z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.
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%I #7 Dec 15 2016 23:43:14

%S 1,3,4,5,9,10,14,16,17,18,19,20,22,24,29,33,34,35,37,41,45,48,49,50,

%T 51,52,53,58,59,61,64,65,66,68,69,70,73,74,77,78,80,82,84,88,89,90,94,

%U 97,98,99,100,104,106,107,109,113,114,116,117,121,122,125,129,130,132,133,138,139,141,144

%N Numbers of the form x^2 + y^2 + z^2 with x + 2*y - 2*z a power of four (including 4^0 = 1), where x,y,z are nonnegative integers.

%C Part (i) of the conjecture in A279612 implies that any positive integer can be written as the sum of a square and a term of the current sequence.

%C It seems that a(n)/n has the limit 2 as n tends to the infinity.

%H Zhi-Wei Sun, <a href="/A279616/b279616.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(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 + 2*0 - 2*0 = 4^0.

%e a(2) = 3 since 3 = 1^2 + 1^2 + 1^2 + 0^2 with 1 + 2*1 - 2*1 = 4^0.

%e a(4) = 5 since 5 = 2^2 + 1^2 + 0^2 + 0^2 with 2 + 2*1 - 2*0 = 4.

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

%t FP[n_]:=FP[n]=n>0&&IntegerQ[Log[4,n]];

%t ex={};Do[Do[If[SQ[m-x^2-y^2]&&FP[x+2y-2*Sqrt[m-x^2-y^2]],ex=Append[ex,m];Goto[aa]],{x,0,Sqrt[m]},{y,0,Sqrt[m-x^2]}];Label[aa];Continue,{m,1,144}]

%Y Cf. A000079, A000290, A271518, A275656, A275675, A275676, A275738, A278560, A279612.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Dec 15 2016