{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Padovan sequence\n",
    "\n",
    "## Try me\n",
    "[![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ffraile/computer_science_tutorials/blob/main/source/Extra%20Exercises/Ex9.%20Padovan%20(Solved).ipynb)[![Binder](https://mybinder.org/badge_logo.svg)](https://mybinder.org/v2/gh/ffraile/computer_science_tutorials/main?labpath=source%2FExtra%20Exercises%2FEx9.%20Padovan%20(Solved).ipynb)\n",
    "\n",
    "The Padovan sequence is the following infinite sequence of natural numbers:\n",
    "`1 1 1 2 2 3 4 5 7 9 12 16 21 28 37 ...`\n",
    "\n",
    "The sequence begins with the subsquence $[1, 1, 1]$ and then each term is the sum of two terms before the last one, that is:\n",
    "\n",
    "$P(0) = 1$\n",
    "\n",
    "$P(1) = 1$\n",
    "\n",
    "$P(2) = 1$\n",
    "\n",
    "$P(n) = P(n-2) + P(n-3)$\n",
    "\n",
    "The Padovan sequence is a cousin of the Fibonacci sequence and has many applications in applied mathematics. As a curiosity, the Padovan sequence can be represented as a set of joined equilateral triangles forming an spiral:\n",
    "\n",
    "<a title=\"The original uploader was Gandalf61 at English Wikipedia., CC BY-SA 3.0 &lt;http://creativecommons.org/licenses/by-sa/3.0/&gt;, via Wikimedia Commons\" href=\"https://commons.wikimedia.org/wiki/File:Padovan_triangles_(1).png\"><img width=\"512\" alt=\"Padovan triangles (1)\" src=\"https://upload.wikimedia.org/wikipedia/commons/c/cd/Padovan_triangles_%281%29.png\"></a>\n",
    "\n",
    "Recall that in Python, the last elements of the array can be accessed directly using negative indices, for example, `array[-1]` would be the last element of the array, `array [-2]` the second to last, and so on. Also, remember that the `append()` function, adds an element to the end of the array.\n",
    "\n",
    "With everything explained so far, **develop the following program**:\n",
    "\n",
    "A program that asks the user for a length and generates a Padovan sequence of that length, stores it in an array and prints the array. "
   ]
  },
  {
   "cell_type": "markdown",
   "source": [
    "# Solution\n",
    "## Flow chart\n",
    "The following flow chart represents a possible solution to the problem:\n",
    "\n",
    "![Padovan Sequence](img/Padovan.png)\n",
    "\n",
    "We can generate a Padovan sequence with 4 elements as:\n",
    "\n",
    "\n",
    "```python\n",
    "pad = [1, 1, 1]\n",
    "next_val = pad[-2] + pad[-3]\n",
    "pad.append(next_val)\n",
    "```\n",
    "\n",
    "Note that the value that we have added does not depend on the length of the array, but on the last two elements of the array. Therefore, we can use a  loop to generate the sequence of any length.\n",
    "\n",
    "Also, in Python, we can use the `while` loop to iterate while a certain condition is not met. We can use this type of loop to generate a Padovan sequence with a  given length, using the ```len``` function in the condition to compare the actual length of the array with the value provided by the user.\n",
    "\n",
    "## Code\n",
    "This is the code of the solution:"
   ],
   "metadata": {
    "collapsed": false
   }
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "outputs": [],
   "source": [
    "# Get length of sequence to generate from input\n",
    "padovan_len = int(input(\"enter length of sequence: \"))\n",
    "# Init Padovan sequence array\n",
    "pad = [1, 1, 1]\n",
    "\n",
    "# While the length of the sequence is smaller than the input\n",
    "while len(pad) < padovan_len:\n",
    "    # Calculate the next value as the sum of the two values before the last one\n",
    "    next_val = pad[-2] + pad[-3]\n",
    "    # Append the new value to the array\n",
    "    pad.append(next_val)\n",
    "\n",
    "print(pad)\n"
   ],
   "metadata": {
    "collapsed": false
   }
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.4"
  },
  "pycharm": {
   "stem_cell": {
    "cell_type": "raw",
    "source": [],
    "metadata": {
     "collapsed": false
    }
   }
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}