Pell Numbers

The Pell numbers are an integer sequence beginning zero, one, two, five, twelve, twenty-nine and so on, in which each term after the second is twice the preceding term plus the one before that. The sequence takes its name from the seventeenth century English mathematician John Pell, though the attribution is itself the result of a historical mix-up, since Leonhard Euler mistakenly credited Pell with work on the related Pell equation that was actually done earlier by other mathematicians, including William Brouncker. The ratio of consecutive Pell numbers converges to one plus the square root of two, a constant sometimes called the silver ratio by analogy with the golden ratio produced by the Fibonacci sequence, and the numbers themselves provide increasingly accurate rational approximations to the square root of two. Pell numbers also appear as the numerators and denominators of the best rational approximations generated by the continued fraction expansion of that same square root.