A364385 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) and the coefficients u, v, w as well as the solutions x_1, x_2 are integers.

1, 4, 6, 12, 15, 21, 23, 31, 35, 42, 46, 54, 56, 66, 70, 78, 83, 93, 95, 105, 109, 119, 123, 133, 137, 148, 154, 162, 166, 178, 180, 194, 198, 206, 212, 224, 229, 241, 245, 255, 259, 273, 275, 289, 295, 303, 309, 321, 325, 340, 346, 356, 360, 372, 376, 390, 396

Offset: 1

Author

Felix Huber, Jul 22 2023

Keywords

• easy
• nonn

Examples

			For n = 3 the a(3) = 6 solutions (u, v, w, x_1, x_2) with positive u are (1, 0, 0, 0, 0), (1, -1, 0, 1, 0), (1, 0, -1, 1, -1), (1, 1, 0, 0, -1), (1, -2, 0, 2, 0), (1, 2, 0, 0, -2).
Equations multiplied by -1 do not have a different solution set, for example (-1, 1, 0, 1, 0) has the same solution set as (1, -1, 0, 1, 0).
		

Links

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

Crossrefs

Partial sums of A364384.

Cf. 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;
A364385 := proc(n) local s; option remember; if n = 1 then A364384(1); else procname(n - 1) + A364384(n); end if; end proc; seq(A364385(n), n = 1 .. 57);

Python

from math import gcd
from sympy import integer_nthroot
def A364385(n):
    c = 0
    for v in range(0,n):
        for w in range(0,n-v):
            for u in range(1,n-v-w+1):
                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 v>0 and v2-w2>=0:
                        d, r = integer_nthroot(v2-w2,2)
                        if r and not((d+v)%u2 or (d-v)%u2):
                            c += 1
                            if w>0:
                                c += 1
    return c # Chai Wah Wu, Oct 04 2023

Formula

a(n) = Sum_{k=1..n} A364384(k).