add problem 65

notebooks/problem0065.ipynb

@@ -0,0 +1,137 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "bdec47c4",
+   "metadata": {},
+   "source": [
+    "# [Convergents of $e$](https://projecteuler.net/problem=65)\n",
+    "\n",
+    "Easy one-liner in SageMath."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "8a93028c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "272"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sum(continued_fraction(e).convergent(99).numerator().digits())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "396468b7",
+   "metadata": {},
+   "source": [
+    "If we wanted to compute the convergents ourselves, we could first make a generator for the partial denominators of the continued fraction of $e$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "bedc34c8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from itertools import count, chain\n",
+    "\n",
+    "def partial_denominators_e(n):\n",
+    "    yield 2\n",
+    "    denominators = chain.from_iterable((1, 2 * k, 1) for k in count(1))\n",
+    "    for (i, b) in enumerate(denominators):\n",
+    "        if i >= n:\n",
+    "            break\n",
+    "            \n",
+    "        yield b"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "33288cd0",
+   "metadata": {},
+   "source": [
+    "Then write a function for computing a continued fraction from a sequence of partial denominators (outside of SageMath, you might want to use a [fraction type](https://docs.python.org/3/library/fractions.html))."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "1f7bab23",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "def cf(denominators):\n",
+    "    a = next(denominators)\n",
+    "    \n",
+    "    try:\n",
+    "        return a + 1 / cf(denominators)\n",
+    "    except StopIteration:\n",
+    "        return a"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "5b8189b7",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "272"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "sum(cf(partial_denominators_e(99)).numerator().digits())"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5778175d",
+   "metadata": {},
+   "source": [
+    "## Relevant sequences\n",
+    "* Numerators of convergents of $e$: [A007676](https://oeis.org/A007676)"
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.5",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.2"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}