The Pell numbers follow \(P_n = 2P_{n-1} + P_{n-2}\) from seeds 0, 1: they run 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378.
Where Fibonacci ratios converge to the golden ratio, consecutive Pell ratios converge to the silver ratio \(1 + \sqrt{2}\). The fractions \(\tfrac{3}{2}, \tfrac{7}{5}, \tfrac{17}{12}, \tfrac{41}{29}\), built from Pell numbers, are the best rational approximations to \(\sqrt{2}\) — the ancient Greeks' "side and diameter numbers" used exactly this recurrence two millennia before Pell.
Pell numbers also solve the classic square–triangular problem: every number that is simultaneously square and triangular (1, 36, 1225, 41616, …) comes from products of consecutive Pell numbers.