A364384 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with different solution sets L != {}, where n = abs(u) + abs(v) + abs(w), the coefficients u, v, w as well as the solutions x_1, x_2 are integers and GCD(u, v, w) = 1.

1, 3, 2, 6, 3, 6, 2, 8, 4, 7, 4, 8, 2, 10, 4, 8, 5, 10, 2, 10, 4, 10, 4, 10, 4, 11, 6, 8, 4, 12, 2, 14, 4, 8, 6, 12, 5, 12, 4, 10, 4, 14, 2, 14, 6, 8, 6, 12, 4, 15, 6, 10, 4, 12, 4, 14, 6, 12, 4, 14, 2, 14, 6, 10, 9, 14, 4, 12, 4, 12, 4, 18, 2, 16, 6, 8, 8, 12, 4, 16, 6, 13, 6, 14, 4, 14, 6, 10, 4, 18, 4, 18, 6, 8, 6, 14, 4, 16, 6, 14

Offset: 1

Author

Felix Huber, Jul 22 2023

Keywords

• easy
• nonn

Examples

			For n = 4 the a(4) = 6 solutions (u, v, w, x_1, x_2) with positive u are (1, -3, 0, 3, 0), (1, -2, 1, 1, 1), (1, -1, -2, 2, -1), (1, 1, -2, 1, -2), (1, 2, 1, -1, -1), (1, 3, 0, 0, -3).
Equations multiplied by -1 do not have a different solution set; for example, (-1, 3, 0, 3, 0) has the same solution set as (1, -3, 0, 3, 0).
Equations with GCD(u, v, w) != 1 are not considered, they belong to a lower n. For example (2, 2, 0, 0, -1) ist not considered here, it belongs to n = 2 with (1, 1, 0, 0, -1).
		

Links

• Felix Huber, Equations for a given n.
• Wikipedia, Quadratic equation.

Crossrefs

Cf. A364385 (partial sums), A365876, A365877, A365892

Programs

Maple

A364384 := proc(n) local i, u, v, w, x_1, x_2, a; a := 0; i := n; for v from 1 - i to i - 1 do for w from abs(v) - i + 1 to i - abs(v) - 1 do u := i - abs(v) - abs(w); if igcd(u, v, w) = 1 then x_1 := 1/2*(-v + sqrt(v^2 - 4*w*u))/u; x_2 := 1/2*(-v - sqrt(v^2 - 4*w*u))/u; if floor(Re(x_1)) = x_1 and floor(Re(x_2)) = x_2 then a := a + 1; end if; end if; end do; end do; end proc; seq(A364384(n), n = 1 .. 100);

Python

from math import gcd
from sympy import integer_nthroot
def A364384(n):
    if n == 1: return 1
    c = 0
    for v in range(0,n):
        for w in range(0,n-v):
            u = n-v-w
            if gcd(u,v,w)==1:
                v2, w2, u2 = v*v, w*(u<<2), u<<1
                if v2+w2>=0:
                    d, r = integer_nthroot(v2+w2,2)
                    if r and not ((d+v)%u2 or (d-v)%u2):
                        c += 1
                        if v>0 and w>0:
                            c += 1
                if v2-w2>=0:
                    d, r = integer_nthroot(v2-w2,2)
                    if r and not((d+v)%u2 or (d-v)%u2):
                        c += 1
                        if v>0 and w>0:
                            c += 1
    return c # Chai Wah Wu, Oct 04 2023