The questions asks:
Define a formula to yield the Kth N for which there exists an integer X less than or equal to N for which the sum of the integers from 0 to X (inclusive) is equal to the sum of the integers from X to N (inclusive). Through a bit of algebra, we can paraphrase the question as: Find the Kth N for which N(N+1)/2 equal to some perfect square.
I have defined a recursive formula for this:
f(1) = 1
f(2) = 8
f(3) = 49
f(k) = 4f(k-2)(f(k-2) + 1) for k > 3
So the numbers are: 1,8,49,288,9800...
My question is if it is possible to find an "implicit" function for yielding N.
The answer to the title question is $f(k)=\dfrac{(3-2\sqrt2)^k+(3+2\sqrt2)^k-2}4$.
For further information, see this related question.
This sequence is in OEIS and includes $1681,$ which your formula doesn't have.