add problem 65

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filifa 2025-05-15 18:19:05 -04:00
parent 4f8da514cc
commit b90b6d1e4c
1 changed files with 137 additions and 0 deletions

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notebooks/problem0065.ipynb Normal file
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{
"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
}