Plus Minus Four Squares

Every non-negative integer $n$ may be written as the sum of the squares of four integers:

$$n = a^2 + b^2 + c^2 + d^2$$

By allowing subtraction, $n$ may be written in many more ways; in fact infinitely many.

In this problem you will count the number of different ways to express an input $n$ as a sum or difference of four squares with several restrictions:

First, we need to decide what different means.

Any of $a$, $b$, $c$, $d$ may be replaced by its negative. We do not want to count these as different so we will only count different squared values.

Reordering $a$, $b$, $c$, $d$ does not give a different representation.

So, we define a plus minus four square representation of a non-negative integer $n$ as a sequence of four perfect squares in non-increasing order with plus or minus signs whose computation results in $n$.

In addition, we add the following restrictions:

• The first square must be no more than $n$ to avoid having infinitely many representations.
• If the same square appears multiple times all appearances must be preceded by (a possibly implicit) plus sign or all must be preceded by a minus sign. This avoids something like: $$64 + 36 - 36 + 0$$
• A square of zero must be preceded by a plus sign.

For example, the only sums of squares which add to 64 are:

$$64 + 0 + 0 + 0$$ $$16 + 16 + 16 + 16$$

If we allow minus signs with the above additional restrictions we have the following which each sum up to $64$:

$$64 + 25 - 16 - 9$$ $$64 - 25 + 16 + 9$$ $$64 + 0 + 0 + 0$$ $$49 + 49 - 25 - 9$$ $$49 + 36 - 25 + 4$$ $$49 + 25 - 9 - 1$$ $$49 + 16 - 1 + 0$$ $$36 + 36 - 9 + 1$$ $$36 + 36 - 4 - 4$$ $$36 + 25 + 4 - 1$$ $$36 + 16 + 16 - 4$$ $$16 + 16 + 16 + 16$$

Write a program which takes as input a non-negative integer $n$ and outputs a count of the number of different four square plus minus representations of $n$.

Input consists of one line containing a single non-negative decimal integer ($0 < n ≤ 5000$).

There is one line of output that consists of a single decimal integer giving a count of the number of different four square plus minus representations of $n$.