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The Pell Numbers[FINALIZED]
#3: Post edited
by
Razetime
·
2021-02-17T04:42:02Z (almost 4 years ago)
finalized
The Pell Numbers
- The Pell Numbers[FINALIZED]
- # 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}&\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 code-golf. Shortest answer in each language wins.
#2: Post edited
by
Razetime
·
2021-02-10T04:10:53Z (almost 4 years ago)
- # 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}&\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}\$ number of the sequence.- 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(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.
#1: Initial revision
by
Razetime
·
2021-02-10T04:09:35Z (almost 4 years ago)
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}&\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}\$ number of the sequence.
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.