I am interested in all solutions in odd positive integers d, e, k, with d<=k and e<=k of the equation d^2 + (k-1)e^2 = k(k^2 + 2).
This came up during work on dominating sets of the queen's graph (finding for each n the minimum number of queens one can place on an n-by-n board so that every square is either occupied or covered by a queen).
For each odd positive k, there is at most one odd positive e giving a solution (k, e, d).
I can see see that there are infinite sequences of solutions, related to Pell-type equations of the form X^2 - 3Y^2 = constant.
The density of solutions appears to decrease as k increases.