{
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  {
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   "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)"
   ]
  }
 ],
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