I've just rephrased the question a bit.
S(i=1 to n) is a set of n whole numbers where Square_Root(Si) is always a whole number.
Di(i=1 to n) is a set of differences such that:
Si+Di = Sj where Square_Root(Sj) is a whole number and there no other whole number between Si and Sj whose square root is a whole number.
For a random value Si, I need to find the Di.
Once I get the Di for a given Si, I can easily find the next value of D in the sequence.
I am a novice in mathematics and am writing a software program. Any help would be greatly appreciated.!
You want to go from $k^2$ to $(k+1)^2$, so you do the following $$ (k+1)^2 - k^2 = 2k+1, $$ and so starting with $a_0=0$ (implies a square of $0^2=0$), the next one would be $$ a_1 = a_0 + 2\cdot 0 + 1 = a_0 + 1 = 1\\ a_2 = a_1 + 2\cdot 1 + 1 = a_1 + 3 = 4\\ a_3 = a_2 + 2\cdot 2 + 1 = a_2 + 5 = 9 $$
This way you can generate the entire sequence.
UPDATE
Starting at any random value $N$ which is a perfect square, compute the square root and set $k=\sqrt{N}$, then proceed as before, with $$ a_{k+1} = a_k+2k+1 $$