Problem 456

Triangles containing the origin II

Define:
x_n = (1248ⁿ mod 32323) - 16161
y_n = (8421ⁿ mod 30103) - 15051
P_n = {(x₁, y₁), (x₂, y₂), …, (x_n, y_n)}

For example, P₈ = {(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)}.

Let C(n) be the number of triangles whose vertices are in P_n which contain the origin in the interior.

Examples:
C(8) = 20
C(600) = 8950634
C(40 000) = 2666610948988

Find C(2 000 000).

包含原点的三角形II

我们定义：
x_n = (1248ⁿ mod 32323) - 16161
y_n = (8421ⁿ mod 30103) - 15051
P_n = {(x₁, y₁), (x₂, y₂), …, (x_n, y_n)}

例如，P₈ = {(-14913, -6630), (-10161, 5625), (5226, 11896), (8340, -10778), (15852, -5203), (-15165, 11295), (-1427, -14495), (12407, 1060)}.

以P_n中元素为顶点的三角形中，包含有原点在其内部的三角形数目记为C(n)。

例如：
C(8) = 20
C(600) = 8950634
C(40 000) = 2666610948988

求C(2 000 000)。