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