Solvability criteria of negative Pell equations x 2 − dy 2 = −1 have previously been established via calculating the length for the period of the simple continued fraction of √ d and checking the existence of a primitive Pythagorean triple for d. However, when d 1, such criteria usually require a lengthy calculation. In this note, we establish a novel approach to construct integers d such that x 2 − dy 2 = −1 is solvable in integers x and y, where d = d(u n , u n+1 , m) can be expressed as rational functions of u n and u n+1 and fourth-degree polynomials of m, and u n satisfies a recurrence relation: u 0 = u 1 = 1 and u n+2 = 3u n+1 − u n for n ∈ N ∪ {0}. Our main argument is based on a binary quadratic relation between u n and u n+1 and properties 1+u 2 n u n+1 ∈ N and 1+u 2 n+1 un ∈ N. Due to the recurrence relation of u n , such d's are easy to be generated by hand calculation and computational mathematics via a class of explicit formulas. Besides, we consider equation x 2 − k(k + 4)m 2 y 2 = −1 and show that it is solvable in integers if and only if k = 1 and m ∈ N is a divisor of 1 2 u 3n+2 for some n ∈ N ∪ {0}. The main approach for its solvability is the Fermat's method of infinite descent.