A169580 Squares of the form x^2 + y^2 + z^2 with x, y, z as 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

1,1

Comments

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

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

References

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

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..3140 from Robert Israel)

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

Formula

For n >= 32, a(n) = (n+m+1)^2 if n+m >= A094958(m+1) and a(n) = (n+m)^2 otherwise, where m = A070939(n)+A070939(floor(n/5)). - Chai Wah Wu, Jan 25 2026

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 49 - 36 - 9 - 4 = 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(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[58]^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 *)

Prog

(Python)
def A169580(n): return (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)[n-1] if n<32 else (n+(m:=n.bit_length()+(n//5).bit_length())+(n+m>=(1<<m if m<3 else 5<<((m+1>>1)-2) if m&1 else 1<<(m+1>>1)+1)))**2 # Chai Wah Wu, Jan 25 2026
(PARI) apply(sqr, select(n->!(if(n%5, n, n/5)==2^valuation(n, 2)), [1..58])) \\ Bruce Nye, May 28 2026

Crossrefs

Cf. A000378, A000419, A005767 (square roots), A070939, A094958.

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