Pell Number Calculator

Formula

The formula to calculate the Pell Number for the Nth term is:

\[ P_n = 2P_{n-1} + P_{n-2} \]

Where:

\( P_n \) is the Pell number at the Nth term
\( P_{n-1} \) is the Pell number at the (N-1)th term
\( P_{n-2} \) is the Pell number at the (N-2)th term

What is a Pell Number?

Pell numbers are a sequence of integers that are solutions to the Pell equation, a specific type of Diophantine equation. The sequence begins with \( P_0 = 0 \) and \( P_1 = 1 \), and each subsequent number is the sum of twice the previous Pell number and the Pell number before that. Pell numbers have applications in number theory and the approximation of square roots.

Example Calculation

Let's assume we want to find the Pell number for the 5th term (N = 5):

\( P_0 = 0 \)
\( P_1 = 1 \)
\( P_2 = 2P_1 + P_0 = 2 \times 1 + 0 = 2 \)
\( P_3 = 2P_2 + P_1 = 2 \times 2 + 1 = 5 \)
\( P_4 = 2P_3 + P_2 = 2 \times 5 + 2 = 12 \)
\( P_5 = 2P_4 + P_3 = 2 \times 12 + 5 = 29 \)

The Pell number for the 5th term is 29.