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Introduction The Pell(no, not Bell) Numbers are a simple, Fibonacci-like sequence, defined by the following relation: $P_n=\begin{cases}0&\mbox{if }n=0;\\1&\mbox{if }n=1;\\2P_{n-1}+P_{n-2...
#2: Post edited
The Pell Numbers
- # Introduction
- The [Pell](https://en.wikipedia.org/wiki/Pell_number)(no, not Bell) Numbers are a simple, Fibonacci-like sequence, defined by the following relation:
- \$P_n=\begin{cases}0&\mbox{if }n=0;\\\\1&\mbox{if }n=1;\\\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}\$
- They also have a closed form:
- \$P_n=\frac{\left(1+\sqrt2\right)^n-\left(1-\sqrt2\right)^n}{2\sqrt2}\$
- And a matrix multiplication based form, for the daring:
- \$\begin{pmatrix} P_{n+1} & P_n \\\\ P_n & P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\\\ 1 & 0 \end{pmatrix}^n.\$
- # Challenge
- Your mission, should you choose to accept it, is to do any one of the following:
- 1. Given \$n\$, calculate the \$n^{th}\$ term of the sequence (0 or 1-indexed).
- 2. Given \$n\$, calculate the first \$n\$ elements of the sequence.
- 3. Output the sequence indefinitely.
- # Scoring
This is code-golf. Shortest answer in each language wins.
- # Introduction
- The [Pell](https://en.wikipedia.org/wiki/Pell_number)(no, not Bell) Numbers are a simple, Fibonacci-like sequence, defined by the following relation:
- \$P_n=\begin{cases}0&\mbox{if }n=0;\\\\1&\mbox{if }n=1;\\\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}\$
- They also have a closed form:
- \$P_n=\frac{\left(1+\sqrt2\right)^n-\left(1-\sqrt2\right)^n}{2\sqrt2}\$
- And a matrix multiplication based form, for the daring:
- \$\begin{pmatrix} P_{n+1} & P_n \\\\ P_n & P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\\\ 1 & 0 \end{pmatrix}^n.\$
- # Challenge
- Your mission, should you choose to accept it, is to do any one of the following:
- 1. Given \$n\$, calculate the \$n^{th}\$ term of the sequence (0 or 1-indexed).
- 2. Given \$n\$, calculate the first \$n\$ elements of the sequence.
- 3. Output the sequence indefinitely.
- # Scoring
- This is <a class="badge is-tag">code-golf</a>. Shortest answer in each language wins.
#1: Initial revision
The Pell Numbers
# Introduction The [Pell](https://en.wikipedia.org/wiki/Pell_number)(no, not Bell) Numbers are a simple, Fibonacci-like sequence, defined by the following relation: \$P_n=\begin{cases}0&\mbox{if }n=0;\\\\1&\mbox{if }n=1;\\\\2P_{n-1}+P_{n-2}&\mbox{otherwise.}\end{cases}\$ They also have a closed form: \$P_n=\frac{\left(1+\sqrt2\right)^n-\left(1-\sqrt2\right)^n}{2\sqrt2}\$ And a matrix multiplication based form, for the daring: \$\begin{pmatrix} P_{n+1} & P_n \\\\ P_n & P_{n-1} \end{pmatrix} = \begin{pmatrix} 2 & 1 \\\\ 1 & 0 \end{pmatrix}^n.\$ # Challenge Your mission, should you choose to accept it, is to do any one of the following: 1. Given \$n\$, calculate the \$n^{th}\$ term of the sequence (0 or 1-indexed). 2. Given \$n\$, calculate the first \$n\$ elements of the sequence. 3. Output the sequence indefinitely. # Scoring This is code-golf. Shortest answer in each language wins.