A279616 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.

1, 3, 4, 5, 9, 10, 14, 16, 17, 18, 19, 20, 22, 24, 29, 33, 34, 35, 37, 41, 45, 48, 49, 50, 51, 52, 53, 58, 59, 61, 64, 65, 66, 68, 69, 70, 73, 74, 77, 78, 80, 82, 84, 88, 89, 90, 94, 97, 98, 99, 100, 104, 106, 107, 109, 113, 114, 116, 117, 121, 122, 125, 129, 130, 132, 133, 138, 139, 141, 144

Offset: 1

Author

Zhi-Wei Sun, Dec 15 2016

Keywords

• nonn

Comments

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.

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

Examples

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

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, arXiv:1604.06723 [math.GM], 2016.

Crossrefs

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

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
FP[n_]:=FP[n]=n>0&&IntegerQ[Log[4,n]];
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}]