what is the set of all positive integers satisfying the equality $2(n^2 - 1)=t(t+1)$ other than $n=t=2$ or $n=4$, $t=5$?
You can rewrite it as $(2t + 1)^2 = 8n^2 - 7$, so it's essentially equivalent to finding solution for the Pell equation $x^2 - 8y^2 = -7$. This has one solution $(1,1)$, and so it certainly has infinitely many. For example, the solution $(x,y) = (379, 134)$ gives you $n = 134, t = 189$. See, for instance, "An introduction to the theory of numbers" by Niven, Zuckerman and Montgomery, Section 7.8.