Right-angled Triangle

A triangle is one of the basic geometric shapes: a polygon with three vertices, not lying on a straight line, and three sides, which are its line segments. A triangle with vertices $\text{A}$, $\text{B}$, and $\text{C}$ is denoted as $\text{△ABC}$.

Triangles can be classified by their internal angles as follows:

• A right-angled triangle has one angle of 90^\circ. The side opposite the right angle is called the hypotenuse and is the longest side. The other two sides are called the legs. For right-angled triangles, the Pythagorean theorem holds: the sum of the squares of the legs equals the square of the hypotenuse, i.e. a^2 + b^2 = c^2, where $a$ and $b$ are the legs and $c$ is the hypotenuse.

• An oblique triangle is a triangle with no interior angle of 90^\circ.

• An obtuse triangle is an oblique triangle with one angle greater than 90^\circ (this is called an obtuse angle).

• An acute triangle is an oblique triangle with all interior angles less than 90^\circ (three acute angles). An equilateral triangle is an example of an acute triangle, but not all acute triangles are equilateral.

Your task is very simple. For a given natural number $L$, count how many different right-angled triangles satisfy a + b + c \leq L, where $a$ and $b$ are the legs and $c$ is the hypotenuse. Note that $a$, $b$, and $c$ must all be integers.