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

%I A169580 #23 Mar 17 2016 08:50:06
%S A169580 9,36,49,81,121,144,169,196,225,289,324,361,441,484,529,576,625,676,
%T A169580 729,784,841,900,961,1089,1156,1225,1296,1369,1444,1521,1681,1764,
%U A169580 1849,1936,2025,2116,2209,2304,2401,2500,2601,2704,2809,2916,3025,3136,3249,3364
%N A169580 Squares of the form x^2+y^2+z^2 with x,y,z positive integers.
%C A169580 Integer solutions of a^2 = b^2 + c^2 + d^2, i.e., Pythagorean Quadruples. - _Jon Perry_, Oct 06 2012
%C A169580 Also null (or light-like, or isotropic) vectors in Minkowski 4-space. - _Jon Perry_, Oct 06 2012
%D A169580 T. Nagell, Introduction to Number Theory, Wiley, 1951, p. 194.
%H A169580 Robert Israel, <a href="/A169580/b169580.txt">Table of n, a(n) for n = 1..3140</a>
%H A169580 O. Fraser and B. Gordon, <a href="http://www.jstor.org/stable/2317949">On representing a square as the sum of three squares</a>, Amer. Math. Monthly, 76 (1969), 922-923.
%H A169580 D. Rabahy, <a href="https://docs.google.com/spreadsheets/d/11Wap63eI6_ELlaPOiELDQzukdqYaGt-TDfD3I5b5F0U/pubhtml">Spreadsheet revealing sequence in table of all a^2+b^2+c^2</a>
%H A169580 David Rabahy, <a href="/A169580/a169580.png">a^2+b^2+c^2 squares colored, zoomed way out to make "lines" apparent</a>
%e A169580 9 = 1 + 4 + 4,
%e A169580 36 = 16 + 16 + 4,
%e A169580 49 = 36 + 9 + 4,
%e A169580 81 = 49 + 16 + 16,
%e A169580 so these are in the sequence.
%e A169580 16 cannot be written as the sum of 3 squares if zero is not allowed, therefore 16 is not in the sequence.
%e A169580 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
%p A169580 M:= 10000: # to get all terms <= M
%p A169580 sort(convert(select(issqr, {seq(seq(seq(x^2 + y^2 + z^2,
%p A169580   z=y..floor(sqrt(M-x^2-y^2))), y=x..floor(sqrt((M-x^2)/2))),
%p A169580 x=1..floor(sqrt(M/3)))}),list)); # _Robert Israel_, Jan 28 2016
%t A169580 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 *)
%Y A169580 For the square roots see A005767. Cf. A000378, A000419.
%Y A169580 Cf. A217554.
%K A169580 nonn
%O A169580 1,1
%A A169580 _N. J. A. Sloane_, Mar 02 2010