

A169580


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


6



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
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OFFSET

1,1


COMMENTS

Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples.  Jon Perry, Oct 06 2012
Also null (or lightlike, or isotropic) vectors in Minkowski 4space.  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), 922923.
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


EXAMPLE

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 493694=0, so (7,6,3,2) is a null vector in the signatures (+,,,) and (,+,+,+).  Jon Perry, Oct 06 2012


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(Mx^2y^2))), y=x..floor(sqrt((Mx^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 *)


CROSSREFS

For the square roots see A005767. Cf. A000378, A000419.
Cf. A217554.
Sequence in context: A192610 A319958 A329808 * A258844 A294952 A068810
Adjacent sequences: A169577 A169578 A169579 * A169581 A169582 A169583


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Mar 02 2010


STATUS

approved



