A217554 -id:A217554 - cp's OEIS frontend

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

N. J. A. Sloane, Mar 02 2010

nonn

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

			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

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

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

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

Cf. A217554.

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

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 *)

A304726 a(n) = n^4 + 4*n^2 + 3.

Original entry on oeis.org

3, 8, 35, 120, 323, 728, 1443, 2600, 4355, 6888, 10403, 15128, 21315, 29240, 39203, 51528, 66563, 84680, 106275, 131768, 161603, 196248, 236195, 281960, 334083, 393128, 459683, 534360, 617795, 710648, 813603, 927368, 1052675, 1190280, 1340963, 1505528, 1684803

Offset: 0

Vincenzo Librandi, May 31 2018

Alternating sum of all points on the fourth row of the Hosoya triangle composed of Fibonacci polynomials, where F_{0}(n) = 1 and F_{1}(n) = n, hence a(n) = F_{5}(n)/F_{1}(n) for n>0 (see Florez et al. reference, page 7, Table 4 and following sum).

Apart from 8, all terms belong to A217554 because a(n) = (n^2+1)^2 + (n+1)^2 + (n-1)^2 = (n^2+2)^2 - 1. - Bruno Berselli, Jun 04 2018

Muniru A Asiru, Table of n, a(n) for n = 0..10000
Rigoberto Florez, Robinson A. Higuita, and Antara Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.5.
Eric Weisstein's World of Mathematics, Fibonacci Polynomial.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A058071, A059100, A217554.

Subsequence of A005563.

List([0..40], n -> (n^2+2)^2-1); # Muniru A Asiru, Jun 03 2018

Magma
```
[n^4+4*n^2+3: n in [0..40]];
```

seq((n^2+2)^2-1,n=0..40); # Muniru A Asiru, Jun 03 2018

Table[n^4 + 4 n^2 + 3, {n, 0, 35}]
LinearRecurrence[{5,-10,10,-5,1},{3,8,35,120,323},40] (* Harvey P. Dale, Mar 04 2021 *)

G.f.: (3 - 7*x + 25*x^2 - 5*x^3 + 8*x^4)/(1-x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
a(n) = A059100(n)^2 - 1.
Sum_{n>=0} 1/a(n) = 1/6 + coth(Pi)*Pi/4 - coth(sqrt(3)*Pi)*Pi/(4*sqrt(3)). - Amiram Eldar, Feb 24 2023

cp's OEIS Frontend

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

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