A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers.

9, 36, 49, 81, 121, 144, 169, 196, 225, 289, 324, 361, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364

Offset: 1

Author

N. J. A. Sloane, Mar 02 2010

Keywords

• nonn

Comments

Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples. - Jon Perry, Oct 06 2012

Also null (or light-like, or isotropic) vectors in Minkowski 4-space. - Jon Perry, Oct 06 2012

Examples

			9 = 1 + 4 + 4,
36 = 16 + 16 + 4,
49 = 36 + 9 + 4,
81 = 49 + 16 + 16,
so these are in the sequence.
16 cannot be written as the sum of 3 squares if zero is not allowed, therefore 16 is not in the sequence.
Also we can see that 49-36-9-4=0, so (7,6,3,2) is a null vector in the signatures (+,-,-,-) and (-,+,+,+). - _Jon Perry_, Oct 06 2012
		

References

• T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.

Links

• Robert Israel, Table of n, a(n) for n = 1..3140
• O. Fraser and B. Gordon, On representing a square as the sum of three squares, Amer. Math. Monthly, 76 (1969), 922-923.
• D. Rabahy, Spreadsheet revealing sequence in table of all a^2+b^2+c^2
• David Rabahy, a^2+b^2+c^2 squares colored, zoomed way out to make "lines" apparent

Crossrefs

For the square roots see A005767. Cf. A000378, A000419.

Cf. A217554.

Programs

Maple

M:= 10000: # to get all terms <= M
sort(convert(select(issqr, {seq(seq(seq(x^2 + y^2 + z^2,
  z=y..floor(sqrt(M-x^2-y^2))), y=x..floor(sqrt((M-x^2)/2))),
x=1..floor(sqrt(M/3)))}),list)); # Robert Israel, Jan 28 2016

Mathematica

Select[Range[60]^2, Resolve@ Exists[{x, y, z}, Reduce[# == x^2 + y^2 + z^2, {x, y, z}, Integers], And[x > 0, y > 0, z > 0]] &] (* Michael De Vlieger, Jan 27 2016 *)