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%I #24 Oct 05 2023 14:20:24
%S 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,
%T 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,
%U 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
%N 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.
%H Felix Huber, <a href="/A364384/a364384.txt">Equations for a given n</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_equation">Quadratic equation</a>.
%e 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).
%e 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).
%e 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).
%p 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);
%o (Python)
%o from math import gcd
%o from sympy import integer_nthroot
%o def A364384(n):
%o if n == 1: return 1
%o c = 0
%o for v in range(0,n):
%o for w in range(0,n-v):
%o u = n-v-w
%o if gcd(u,v,w)==1:
%o v2, w2, u2 = v*v, w*(u<<2), u<<1
%o if v2+w2>=0:
%o d, r = integer_nthroot(v2+w2,2)
%o if r and not ((d+v)%u2 or (d-v)%u2):
%o c += 1
%o if v>0 and w>0:
%o c += 1
%o if v2-w2>=0:
%o d, r = integer_nthroot(v2-w2,2)
%o if r and not((d+v)%u2 or (d-v)%u2):
%o c += 1
%o if v>0 and w>0:
%o c += 1
%o return c # _Chai Wah Wu_, Oct 04 2023
%Y Cf. A364385 (partial sums), A365876, A365877, A365892
%K easy,nonn
%O 1,2
%A _Felix Huber_, Jul 22 2023
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