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  {
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   "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": [
    "To compute the convergents ourselves, we'll 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",
    "    yield 2\n",
    "    yield from chain.from_iterable((1, 2 * k, 1) for k in count(1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33288cd0",
   "metadata": {},
   "source": [
    "Then we'll apply a simple algorithm for computing [convergents using the partial denominators](https://en.wikipedia.org/wiki/Simple_continued_fraction) (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 convergents(partial_denoms):\n",
    "    h, hprev = 1, 0\n",
    "    k, kprev = 0, 1\n",
    "    for b in partial_denoms:\n",
    "        h, hprev = b * h + hprev, h\n",
    "        k, kprev = b * k + kprev, k\n",
    "        yield h/k"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "99b790f3",
   "metadata": {},
   "source": [
    "Now just iterate until we reach the 100th convergent."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "5b8189b7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "272"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for (i, c) in enumerate(convergents(partial_denominators_e())):\n",
    "    if i == 99:\n",
    "        break\n",
    "\n",
    "sum(c.numerator().digits())"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5778175d",
   "metadata": {},
   "source": [
    "## Relevant sequences\n",
    "* Numerators of convergents of $e$: [A007676](https://oeis.org/A007676)\n",
    "\n",
    "#### Copyright (C) 2025 filifa\n",
    "\n",
    "This work is licensed under the [Creative Commons Attribution-ShareAlike 4.0 International license](https://creativecommons.org/licenses/by-sa/4.0/) and the [BSD Zero Clause license](https://spdx.org/licenses/0BSD.html)."
   ]
  }
 ],
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