Editing Pell number (section)

== Pell numbers ==

The Pell numbers are defined by the [[recurrence relation]]:
:<math>P_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}</math>

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, plus the Pell number before that. The first few terms of the sequence are
:0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, … {{OEIS|id=A000129}}.

Analogously to the [[Binet formula]], the Pell numbers can also be expressed by the closed form formula

<math>P_n=\frac{\left(1+\sqrt2\right)^n-\left(1-\sqrt2\right)^n}{2\sqrt2}.</math>
For large values of ''n'', the {{nowrap|(1 + {{sqrt|2}})<sup>''n''</sup>}} term dominates this expression, so the Pell numbers are approximately proportional to powers of the [[silver ratio]] {{nowrap|1 + {{sqrt|2}}}}, analogous to the growth rate of Fibonacci numbers as powers of the [[golden ratio]].

A third definition is possible, from the [[Matrix (mathematics)|matrix]] formula
:<math>\begin{pmatrix} P_{n+1} & P_n \\ P_n & P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 0 \end{pmatrix}^n.</math>

Many [[identity (mathematics)|identities]] can be derived or [[mathematical proof|proven]] from these definitions; for instance an identity analogous to [[Cassini's identity]] for Fibonacci numbers,
:<math>P_{n+1}P_{n-1}-P_n^2 = (-1)^n,</math>
is an immediate consequence of the matrix formula (found by considering the [[determinant]]s of the matrices on the left and right sides of the matrix formula).<ref>For the matrix formula and its consequences see Ercolano (1979) and Kilic and Tasci (2005). Additional identities for the Pell numbers are listed by Horadam (1971) and Bicknell (1975).</ref>