Pell Numbers
The Pell numbers are defined by the recurrence relation
In words, the sequence of Pell numbers starts with 0 and 1, and then each Pell number is the sum of twice the previous Pell number and the Pell number before that. The first few terms of the sequence are
- 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378... (sequence A000129 in OEIS).
The Pell numbers can also be expressed by the closed form formula
For large values of n, the term dominates this expression, so the Pell numbers are approximately proportional to powers of the silver ratio, analogous to the growth rate of Fibonacci numbers as powers of the golden ratio.
A third definition is possible, from the matrix formula
Many identities can be derived or proven from these definitions; for instance an identity analogous to Cassini's identity for Fibonacci numbers,
is an immediate consequence of the matrix formula (found by considering the determinants of the matrices on the left and right sides of the matrix formula).
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