A364385 -id:A364385 - cp's OEIS frontend

A364384 a(n) is the number of quadratic equations ux^2 + vx + 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

Felix Huber, Jul 22 2023

			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).

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

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

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);

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

A365876 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {}, integer coefficients u, v, w and GCD(u, v, w) = 1, where n = abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 2, 6, 2, 7, 3, 5, 4, 9, 3, 10, 5, 7, 5, 12, 5, 11, 6, 10, 7, 16, 4, 17, 9, 11, 8, 14, 7, 20, 10, 13, 9, 22, 7, 23, 11, 13, 12, 26, 9, 24, 11, 18, 13, 29, 10, 22, 14, 20, 15, 32, 9, 33, 16, 20, 18, 27, 11, 37, 18, 25, 13, 39, 13, 40, 20

Offset: 1

Felix Huber, Sep 22 2023

nonn

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.

			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.

Wikipedia, Quadratic equation.
Wikipedia, Vieta's formulas.

Cf. A364384, A364385, A365877 (partial sums), A365892.

A365876:= proc(n) local u, v, a, min; u := n; v := 0; a := 0; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then a := a + 1; end if; u := u - 2; v:=(n-abs(u))/2; if u < -1/9*n then min := false; end if; end do; return a; end proc; seq(A365876(n), n = 1 .. 74);

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

A365877 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 11, 15, 17, 23, 25, 32, 35, 40, 44, 53, 56, 66, 71, 78, 83, 95, 100, 111, 117, 127, 134, 150, 154, 171, 180, 191, 199, 213, 220, 240, 250, 263, 272, 294, 301, 324, 335, 348, 360, 386, 395, 419, 430, 448, 461, 490, 500, 522, 536, 556, 571, 603

Offset: 1

Felix Huber, Sep 22 2023

nonn

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.

			For n = 11 the a(11) = 23 equations are given by (u, v, w) = (1, 0, 0), (1, 1, -1), (2, 1, -1), (1, 2, -2), (3, 1, -1), (4, 1, -1), (5, 1, -1), (3, 2, -2), (1, 3, -3), (6, 1, -1), (2, 3, -3), (7, 1, -1), (5, 2, -2), (1, 4, -4), (-1, 4, -4), (8, 1, -1), (4, 3, -3), (9, 1, -1), (7, 2, -2), (5, 3, -3), (3, 4, -4), (1, 5, -5), (-1, 5, -5).
Equations multiplied by -1 do not have a different solution set; for example, (- 1, -1, 1) has the same solution set as (1, 1, -1).
Equations with GCD(u, v, w) != 1 are excluded, because their solution set are assigned to equations with lower n. For example, (2, 0, 0) is not included here, because its solution set is already assigned to (1, 0, 0).
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.

Wikipedia, Quadratic equation.
Wikipedia, Vieta's formulas.

Cf. A364384, A364385, A341123, A365892.

Partial sums of A365876.

A365876:= proc(n) local u, v, a, min; u := n; v := 0; a := 0; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then a := a + 1; end if; u := u - 2; v:=(n-abs(u))/2; if u < -1/9*n then min := false; end if; end do; return a; end proc;
A365877:= proc(n) local s; option remember; if n = 1 then A365876(1); else procname(n - 1) + A365876(n); end if; end proc; seq(A365877(n), n = 1 .. 59);

from math import gcd
def A365877(n):
    if n == 1: return 1
    c = 1
    for m in range(2,n+1):
        for v in range(1,m+1>>1):
            u = m-(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 05 2023

a(n) = Sum_{k=1..n} A365876(k).

a(n) = A341123(n) for 1 <= n <= 13.

A365892 a(n) is the index of the n-th term of A365876 that includes at least one equation with at least one integer solution.

Original entry on oeis.org

1, 4, 9, 11, 20, 22, 35, 37, 54, 56, 77, 79, 104, 106, 135, 137, 170, 172, 209, 211, 252, 254, 299, 301, 350, 352, 405, 407, 464, 466, 527, 529, 594, 596, 665, 667, 740, 742, 819, 821, 902, 904, 989, 991, 1080, 1082, 1175, 1177, 1274, 1276, 1377, 1379, 1484, 1486

Offset: 1

Felix Huber, Sep 22 2023

nonn

Observation (checked up to a(52)): a(n) = A266257(n) for n >= 2.

Conjectures in formula section hold for 2<=n<=300. - Chai Wah Wu, Oct 05 2023

			a(1) = 1 since the equation x^2 = 0 belonging to A365876(1) has the integer solution 0. 1 is the 1st term that includes at least one equation with at least one integer solution.
a(2) = 4 since the equation 2*x^2 + x - 1 = 0 belonging to A365876(4) has the integer solution -1. 4 is the 2nd term that includes at least one equation with at least one integer solution.
a(3) = 9 since the equation -x^2 + 4*x - 1 = 0 belonging to A365876(9) has the integer solution 2. 9 is the 3rd term that includes at least one equation with at least one integer solution.
a(4) = 11 since the equation 3*x^2 + 4*x - 4 = 0 belonging to A365876(11) has the integer solution -2. 11 is the 4th term that includes at least one equation with at least one integer solution.

Cf. A364384, A364385, A365876, A365877.

A365892 := proc(n_A365876) local u, v, a, min, x_1, x_2; u := n_A365876; v := 0; a := false; min := true; while min = true do if u <> 0 and gcd(u, v) = 1 then x_1 := 1/2*(-v + sqrt(v^2 + 4*v*u))/u; x_2 := 1/2*(-v - sqrt(v^2 + 4*v*u))/u; if x_1 = floor(x_1) or x_2 = floor(x_2) then a := true; end if; end if; u := u - 2; v := 1/2*n_A365876 - 1/2*abs(u); if u < -1/9*n_A365876 then min := false; end if; end do; if a = true then return n_A365876; end if; end proc; seq(A365892(n_A365876), n_A365876 = 1 .. 1486);

from math import gcd
from itertools import count, islice
from sympy import integer_nthroot
def A365892_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        if n == 1:
            yield 1
        else:
            for v in range(1,n+1>>1):
                u = n-(v<<1)
                if gcd(u,v)==1:
                    v2, u2, a = v*v, v*(u<<2), u<<1
                    if v2+u2 >= 0:
                        d,r = integer_nthroot(v2+u2,2)
                        if r and not ((d+v)%a and (d-v)%a):
                            yield n
                            break
                    if v2-u2 >= 0:
                        d,r = integer_nthroot(v2-u2,2)
                        if r and not ((d+v)%a and (d-v)%a):
                            yield n
                            break
A365892_list = list(islice(A365892_gen(),20)) # Chai Wah Wu, Oct 04 2023

Conjectures (see also A266257): (Start)
a(1) = 1, a(n) = ((n + 1)^2 - (-1)^n*(n - 1))/2 for n >= 2.
a(1) = 1, a(2) = 4, a(3) = 9, a(4) = 11, a(5) = 20, a(6) = 22, a(n) = a(n - 1) + 2*a(n - 2) - 2*a(n - 3) - a(n - 4) + a(n - 5) for n >= 7.
G.f.: (1 + x + 3*x^3 - x^4)/((1 - x)^3*(1 + x)^2). (End)

A373995 Zeros x1 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

Original entry on oeis.org

9, 15, 18, 21, 24, 15, 30, 27, 48, 45, 36, 42, 33, 48, 21, 30, 60, 45, 72, 39, 75, 54, 63, 72, 48, 99, 27, 96, 63, 45, 90, 105, 72, 84, 105, 66, 96, 42, 117, 51, 81, 60, 120, 33, 96, 90, 105, 144, 120, 135, 57, 144, 99, 168, 78, 135, 75, 150, 120, 108, 126, 99, 144

Offset: 1

Felix Huber, Jun 24 2024

nonn

The corresponding values x2 are in A373996. The corresponding maximum values for k, for which the y-coordinates of the extreme points and the inflection point are integers, are in A373997.
These polynomial functions can be used in math lessons when discussing curves. Zeros, extreme points and inflection points can be determined without unnecessary calculation effort with fractions and roots.
Of course, these functions can be stretched in the y-direction by a factor 1/k without affecting the zeros, the extreme points and the inflection point, or shifted in the x-direction, whereby the zeros, the extreme points and the inflection point are also shifted.

			9 is in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 9)*(x - 24) are 4, 18 and 11.
24 is in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 24)*(x - 45) are 10, 36 and 23.

Cf. A373996 (values x2), A373997 (maximum values for k), A364384, A364385.

A373995:=proc(s)
  local x_1,x_2,x_3,x_4,L;
  L:=[];
  for x_1 from 1 to floor((s-1)/2) do
    x_2:=s-x_1;
    x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    if x_3=floor(x_3) and x_4=floor(x_4) then
      L:=[op(L),x_1];
    fi;
  od;
  return op(L);
end proc;
seq(A373995(s),s=3..414);

x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
k = GCD(f(x3), f(x4), f(x5)).

Data corrected by Felix Huber, Aug 18 2024

A373996 Zeros x2 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

Original entry on oeis.org

24, 24, 48, 45, 45, 63, 48, 72, 63, 72, 96, 90, 105, 90, 120, 126, 96, 120, 105, 144, 120, 144, 135, 135, 165, 120, 195, 126, 168, 189, 144, 144, 192, 180, 168, 210, 180, 240, 165, 240, 216, 252, 192, 288, 231, 240, 225, 189, 225, 216, 297, 210, 264, 195, 288, 231, 315, 240, 273, 288, 270, 315, 270

Offset: 1

Felix Huber, Jul 07 2024

nonn

The corresponding values x1 are in A373995. The corresponding maximum values for k, for which the y-coordinates of the extrema and the inflection are integers, are in A373997.
These polynomial functions can be used in math lessons when discussing curves. Zeros, extreme points and inflection points can be determined without unnecessary calculation effort with fractions and roots.
Of course, these functions can be stretched in the y-direction by a factor 1/k without affecting the zeros, the extreme points and the inflection point, or shifted in the x-direction, whereby the zeros, the extreme points and the inflection point are also shifted.

			24 is twice in the sequence, since the x-cordinates of the extreme points and of the inflection point of f(x) = 1/k*x*(x - 9)*(x - 24) are 4, 18 and 11 and of f(x) = 1/k*x*(x - 15)*(x - 24) are 6, 20 and 13.

Cf. A373995 (values x1), A373997 (maximum values for k), A364384, A364385.

A373996:=proc(s)
  local x_1,x_2,x_3,x_4,L;
  L:=[];
  for x_1 from 1 to floor((s-1)/2) do
    x_2:=s-x_1;
    x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    if x_3=floor(x_3) and x_4=floor(x_4) then
      L:=[op(L),x_2];
    fi;
  od;
  return op(L);
end proc;
seq(A373996(s),s=3..414);

x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
k = GCD(f(x3), f(x4), f(x5)).

Data corrected by Felix Huber, Aug 18 2024

A373997 Greatest positive integer k for which the y-coordinates of the extreme points and the inflection point of y = f(x) = 1/k(x - A373995(n))(x - A373996(n)) are integers.

Original entry on oeis.org

2, 2, 16, 2, 2, 2, 16, 54, 2, 54, 128, 16, 2, 16, 2, 16, 128, 250, 2, 2, 250, 432, 54, 54, 2, 2, 2, 16, 686, 54, 432, 2, 1024, 128, 686, 16, 128, 16, 2, 2, 1458, 128, 1024, 2, 2, 2000, 250, 54, 250, 1458, 2, 16, 2662, 2, 16, 2, 250, 2000, 2, 3456, 432, 54, 432

Offset: 1

Felix Huber, Jul 07 2024

nonn

			a(3) = 16, since y = f(x) = 1/16*(x - 18)*(x - 48) has the extrema (8, 200), (36, -486) and the inflection point (22, -143). Since GCD(200, -143, -486) = 1, there is no value of k > 16, for which the y-coordinates of these three points are all integers.

Cf. A373995 (values x1), A373996 (values x2), A364384, A364385.

A373997:=proc(s)
  local x_1,x_2,x_3,x_4,x_5,L;
  L:=[];
  for x_1 from 1 to floor((s-1)/2) do
    x_2:=s-x_1;
    x_3:=(x_1+x_2+sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    x_4:=(x_1+x_2-sqrt(x_1^2+x_2^2-x_1*x_2))/3;
    if x_3=floor(x_3) and x_4=floor(x_4) then
      x_5:=(x_3+x_4)/2;
      L:=[op(L),gcd(gcd(x_3*(x_3-x_1)*(x_3-x_2), x_4*(x_4-x_1)*(x_4-x_2)), x_5*(x_5-x_1)*(x_5-x_2))];
    fi;
  od;
  return op(L);
end proc;
seq(A373997(s),s=3..414);

x-coordinate of the 1. extreme point: x3 = (x1 + x2 + sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the 2. extreme point: x4 = (x1 + x2 - sqrt(x1^2 + x2^2 - x1*x2))/3.
x-coordinate of the inflection point: x5 = (x1 + x2)/3 = (x3 + x4)/2.
k = GCD(f(x3), f(x4), f(x5)).

Data corrected by Felix Huber, Aug 18 2024

A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

Original entry on oeis.org

1, 4, 12, 24, 50, 80, 134, 192, 276, 366, 510, 632, 834, 1018, 1262, 1502, 1858, 2136, 2584, 2956, 3448, 3910, 4576, 5076, 5834, 6488, 7320, 8066, 9136, 9892, 11118, 12114, 13358, 14482, 15978, 17108, 18862, 20272, 22024, 23532, 25700, 27216, 29600, 31486, 33746

Offset: 1

Felix Huber, Feb 18 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and nonnegative discriminant v^2 - 4*u*w have two real solutions.

a(n) is even for n >= 2.

			a(3) = 12 because there are 12 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a nonnegative discriminant: (u, v, w) = (1, 0, 0), (1, -1, 0), (1, 1, 0), (1, 0, -1), (1, -1, -1), (1, 1, -1), (1, -2, 0), (1, 2, 0), (1, 0, -2), (2, -1, 0), (2, 1, 0), and (2, 0, -1). Multiplied equations like 2*(1, 0, 0) = (2, 0, 0) or (-1)*(1, -1, 0) = (-1, 1, 0) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..3800
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A381710, A381711.

A379597:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;	
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 then
               if v=0 then
                  a:=a+1
               elif w=0 or w>=v^2/(4*u) then
                  a:=a+2
               else
                  a:=a+4
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A379597(n),n=1..45);

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.

Original entry on oeis.org

0, 1, 5, 11, 25, 39, 69, 99, 143, 189, 265, 327, 437, 529, 653, 777, 965, 1107, 1343, 1531, 1783, 2021, 2367, 2619, 3013, 3343, 3771, 4153, 4707, 5087, 5721, 6229, 6865, 7437, 8197, 8767, 9677, 10391, 11279, 12043, 13155, 13919, 15147, 16101, 17249, 18301, 19763

Offset: 1

Felix Huber, Mar 06 2025

nonn

Quadratic equations u*x^2 + v*x + w = 0 with real coefficients u, v, w and negative discriminant v^2 - 4*u*w have two complex solutions.

a(n) is odd for n >= 2.

			a(3) = 5 because there are 5 equations with abs(u) + abs(v) + abs(w) <= 3 and distinct solution set having a negative discriminant: (u, v, w) = (1, 0, 1), (1, -1, 1), (1, 1, 1), (1, 0, 2), (2, 0, 1). Multiplied equations like (-1)*(1, -1, 1) = (-1, 1, -1) do not have a distinct solution set.

Felix Huber, Table of n, a(n) for n = 1..5000
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381711.

A381710:=proc(n)
   option remember;
   local a,u,v,w;
      if n=1 then
      0
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
               if igcd(u,v,w)=1 then
                  if v=0 then
                     a:=a+1
                  elif w>v^2/(4*u) then
                    a:=a+2
               fi
            fi
         od
      od;
      a+procname(n-1)
   fi;
end proc;
seq(A381710(n),n=1..47);

A381711 a(n) = A379597(n) - A381710(n).

Original entry on oeis.org

1, 3, 7, 13, 25, 41, 65, 93, 133, 177, 245, 305, 397, 489, 609, 725, 893, 1029, 1241, 1425, 1665, 1889, 2209, 2457, 2821, 3145, 3549, 3913, 4429, 4805, 5397, 5885, 6493, 7045, 7781, 8341, 9185, 9881, 10745, 11489, 12545, 13297, 14453, 15385, 16497, 17517, 18917

Offset: 1

Felix Huber, Mar 08 2025

nonn

a(n) is odd.

Felix Huber, Table of n, a(n) for n = 1..4051
Eric Weisstein's World of Mathematics, Quadratic Equation

Cf. A067274, A091626, A091627, A364384, A364385, A365876, A365877, A379597, A381710.

A381711:=proc(n)
   option remember;
   local a,u,v,w;
   if n=1 then
      1
   else
      a:=0;
      for u to n-1 do
         for v from 0 to n-u do
            w:=n-u-v;
            if igcd(u,v,w)=1 and v<>0 then
               if w=0 or w=v^2/(4*u) then
                  a:=a+2
               elif wA381711(n),n=1..47);

a(n) = A379597(n) - A381710(n).

cp's OEIS Frontend

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.

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A365876 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with distinct solution sets L != {}, integer coefficients u, v, w and GCD(u, v, w) = 1, where n = abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

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A365877 a(n) is the number of quadratic equations u*x^2 + v*x + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

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A365892 a(n) is the index of the n-th term of A365876 that includes at least one equation with at least one integer solution.

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A373995 Zeros x1 of polynomial functions f(x) = 1/k*x*(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

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A373996 Zeros x2 of polynomial functions f(x) = 1/k*x*(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

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A373997 Greatest positive integer k for which the y-coordinates of the extreme points and the inflection point of y = f(x) = 1/k*(x - A373995(n))*(x - A373996(n)) are integers.

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A379597 a(n) is the number of distinct solution sets to the quadratic equations u*x^2 + v*x + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

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A364384 a(n) is the number of quadratic equations ux^2 + vx + 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.

A365876 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {}, integer coefficients u, v, w and GCD(u, v, w) = 1, where n = abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

A365877 a(n) is the number of quadratic equations ux^2 + vx + w = 0 with distinct solution sets L != {} and integer coefficients u, v, w, where n >= abs(u) + abs(v) + abs(w) and the sum of the solutions equals the product of the solutions.

A373995 Zeros x1 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

A373996 Zeros x2 of polynomial functions f(x) = 1/kx(x - x1)*(x - x2), which have three integer zeros 0, x1 and x2 (with 0 < x1 < x2) as well as two extreme points and one inflection point with integer x-coordinates (sorted in ascending order, first by the sum x1 + x2 and then by x1).

A373997 Greatest positive integer k for which the y-coordinates of the extreme points and the inflection point of y = f(x) = 1/k(x - A373995(n))(x - A373996(n)) are integers.

A379597 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a nonnegative discriminant.

A381710 a(n) is the number of distinct solution sets to the quadratic equations ux^2 + vx + w = 0 with integer coefficients u, v, w, abs(u) + abs(v) + abs(w) <= n having a negative discriminant.