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

1,2

Links

Table of n, a(n) for n=1..57.

Felix Huber, Equations for a given n.

Wikipedia, Quadratic equation.

Formula

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

Example

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).

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);

Prog

(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

Crossrefs

Partial sums of A364384.

Cf. A365876, A365877, A365892

Sequence in context: A380230 A251630 A256241 * A247632 A104236 A265225

Adjacent sequences: A364382 A364383 A364384 * A364386 A364387 A364388

Keyword

easy,nonn

Author

Felix Huber, Jul 22 2023

Status

approved