OFFSET
1,5
COMMENTS
According to Vieta's formulas, x_1 + x_2 = -v/u and x_1*x_2 = w/u. So x_1 + x_2 = x_1*x_2 when v = -w. Furthermore, the discriminant must not be negative, i.e., v^2 - 4*u*w = v^2 + 4*u*v >= 0.
LINKS
EXAMPLE
For n = 9 the a(9) = 4 equations are given by (u, v, w) = (7, 1, -1), (5, 2, -2), (1, 4, -4), (-1, 4, -4).
Equations multiplied by -1 do not have a different solution set; for example, (-7, -1, 1) has the same solution set as (7, 1, -1).
Equations with GCD(u, v, w) != 1 are excluded, because their solution sets are assigned to equations with lower n. For example, (3, 3, -3) is not included here, because its solution set is already assigned to (1, 1, -1).
Equations with a double solution are considered to have two equal solutions. For example, (-1, 4, -4) has the two solutions x_1 = x_2 = 2.
MAPLE
PROG
(Python)
from math import gcd
def A365876(n):
if n == 1: return 1
c = 0
for v in range(1, n+1>>1):
u = n-(v<<1)
if gcd(u, v)==1:
v2, u2 = v*v, v*(u<<2)
if v2+u2 >= 0:
c +=1
if v2-u2 >= 0:
c +=1
return c # Chai Wah Wu, Oct 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Sep 22 2023
STATUS
approved