Pell numbers

Example

The Pell numbers $P_n$ are the solution of $$P_{n+2} = 2 P_{n+1} + P_n \qquad (n \ge 0)$$ with $P_0 = 0$ and $P_1 = 1$. The first terms are $0, 1, 2, 5, 12, 29, 70, 169, \dots$. The characteristic polynomial $X^2 - 2X - 1$ has roots $1 \pm \sqrt{2}$, which gives $$P_n = \frac{(1 + \sqrt{2})^n - (1 - \sqrt{2})^n}{2\sqrt{2}}.$$ Check: the formula solves the recurrence, since each root $q$ satisfies $q^2 = 2q + 1$; it gives $P_0 = 0$ and $P_1 = 2\sqrt{2}/(2\sqrt{2}) = 1$. They form the Lucas sequence $U_n(2, -1)$.

Let $H_n$ be the solution of the same recurrence with $H_0 = H_1 = 1$, so $H_n = 1, 1, 3, 7, 17, 41, 99, \dots$. Then $H_n^2 - 2 P_n^2 = (-1)^n$. For example $3^2 - 2 \cdot 2^2 = 1$ and $7^2 - 2 \cdot 5^2 = -1$. So the pairs $(H_n, P_n)$ give solutions of the Pell equations $x^2 - 2y^2 = \pm 1$, and $H_n / P_n$ tends to $\sqrt{2}$.