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Challenges

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Challenges The Pell Numbers

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...

9 answers  ·  posted 3y ago by Razetime‭  ·  last activity 2y ago by torres‭

#2: Post edited by user avatar General Sebast1an‭ · 2021-08-12T12:32:51Z (about 3 years ago)
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 by user avatar Razetime‭ · 2021-02-17T04:41:09Z (over 3 years ago)
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.