117 lines
2.7 KiB
Plaintext
117 lines
2.7 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Convergents of e (Euler Problem 65)\n",
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"\n",
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"Hence the sequence of the first ten convergents for √2 are:\n",
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"\n",
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"1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...\n",
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"\n",
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"What is most surprising is that the important mathematical constant,\n",
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"\n",
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"e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].\n",
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"\n",
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"The first ten terms in the sequence of convergents for e are:\n",
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"\n",
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"2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...\n",
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"\n",
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"The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.\n",
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"\n",
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"Find the sum of digits in the numerator of the 100th convergent of the continued fraction for e."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [],
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"source": [
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"def gcd(a, b):\n",
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" if b > a:\n",
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" a, b = b, a\n",
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" while a % b != 0:\n",
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" a, b = b, a % b\n",
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" return b\n",
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"\n",
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"def add_fractions(n1, d1, n2, d2):\n",
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" d = d1 * d2\n",
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" n1 = n1 * (d // d1)\n",
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" n2 = n2 * (d // d2)\n",
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" n = n1 + n2\n",
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" p = gcd(n, d)\n",
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" return (n // p, d // p)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"272\n"
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]
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}
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],
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"source": [
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"def next_expansion(previous_numerator, previous_denumerator, value):\n",
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" if previous_numerator == 0:\n",
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" return (value, 1)\n",
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" return add_fractions(previous_denumerator, previous_numerator, value, 1)\n",
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"\n",
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"e_sequence = [2] + [n for i in range(2, 1000, 2) for n in (1, i, 1)]\n",
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"\n",
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"n, d = 0, 1\n",
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"\n",
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"for i in range(100, 0, -1):\n",
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" n, d = next_expansion(n, d, e_sequence[i - 1])\n",
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"\n",
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"s = sum([int(l) for l in str(n)])\n",
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"print(s)\n",
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"assert(s == 272)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"completion_date": "Wed, 23 Jan 2019, 05:54",
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"kernelspec": {
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"display_name": "Python 3",
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"language": "python3.6",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.5"
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},
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"tags": [
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"expansion",
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"e",
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"sequence"
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]
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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