Determine the natural number solutions to the equation below

Find all the natural number solutions to the equation $a^{2} + a + 100 = n^{2}$ .

List all $N$ solutions $(a, n)$ to the above equation where $a,n$ belong to the set of natural numbers.

Find the sum of all such ordered pairs $(a,n)$ , i.e., find $\displaystyle\sum_{i=1}^{N}(a_{i} + n_{i})$ .

Note:

In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers {1,2,3,…} according to the traditional definition;

or

the set of non-negative integers {0,1,2,…} according to a definition first appearing in the nineteenth century.

In this problem we need to use the nineteenth century definition..