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  {
   "cell_type": "markdown",
   "id": "552c0f62-f482-4262-8860-32dfc4dcd60c",
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   "source": [
    "# Fibonacci numbers"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2dd0ca2e-e59a-4deb-a033-433656546fa1",
   "metadata": {},
   "source": [
    "\n",
    "In mathematics, the _Fibonacci sequence_ is a sequence in which each element is the sum of the two elements that precede it. \n",
    "Numbers that are part of the Fibonacci sequence are known as **Fibonacci numbers**, commonly denoted $F_n$ "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76d1f24a-cfd8-4377-9a6b-996bab91872d",
   "metadata": {},
   "source": [
    "\n",
    "The Fibonacci numbers may be defined by the *recurrence relation*\n",
    "$F_1 = 1$, $F_2 = 1$, and $F_{n} = F_{n-1} + F_{n-2}$ for $n > 2$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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    "tags": []
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   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    myfib(n)\n",
    "\n",
    "Calculate Fibonacci number F_n using the recurrence relation\n",
    "F_1 = 1, F_2 = 1, and F_n = F_n-1 + F_n-2 for n > 2.\n",
    "\"\"\"\n",
    "function myfib(n)\n",
    "    fp = 1\n",
    "    if n == 1\n",
    "        return fp\n",
    "    end\n",
    "\n",
    "    fc = 1\n",
    "    if n == 2\n",
    "        return fc\n",
    "    end\n",
    "\n",
    "    for i = 3:n\n",
    "        fc, fp = fc + fp, fc\n",
    "    end\n",
    "\n",
    "    return fc\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "2f25f061-5178-42a0-8bba-f5696e273e8b",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "?myfib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6f8e8ed5-09ef-4e61-9534-f9ef1141338a",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "myfib.(1:9)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "284e2246-b9b0-4709-a877-004efd0d751c",
   "metadata": {},
   "source": [
    "\n",
    "A *recursive function* is a programming technique where a function calls itself to solve a problem, breaking it down into simpler, smaller instances of the same problem. \n",
    "It requires a base case, which is a condition that stops the function from calling itself indefinitely, and a recursive step, where the function calls itself with modified \n",
    "input to move closer to the base case. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "eb6ca272-6628-4caf-8499-5bda294d8b28",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\"\"\"\n",
    "    myfib_recursive(n)\n",
    "\n",
    "Calculate Fibonacci number F_n using the _recurrence relation_\n",
    "F_1 = 1, F_2 = 1, and F_n = F_n-1 + F_n-2 for n > 2, \n",
    "and recursive function calls. \n",
    "\"\"\"\n",
    "function myfib_recursive(n)\n",
    "\n",
    "    # Base case\n",
    "    if n == 1\n",
    "        return 1\n",
    "    elseif n == 2\n",
    "        return 1\n",
    "    end\n",
    "\n",
    "    # Recursive step\n",
    "    return myfib_recursive(n-1) + myfib_recursive(n-2)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "af60f5a6-b4c4-47a0-90c7-145681b188d1",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "myfib_recursive.(1:9)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d563be2b-fdc4-4d6a-a65a-db80ec1916a4",
   "metadata": {},
   "source": [
    "### Performance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "851b5c85-0210-4946-9ae9-080a06bbb0cb",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@time myfib(50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "432614d7-c919-4844-97da-8e0781c06a50",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "@time myfib_recursive(50)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6954104b-a8b3-415b-be0a-9b55561f25d3",
   "metadata": {},
   "source": [
    "### Testing"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d8e2fc9b-05d1-46c8-a08d-be269f486b50",
   "metadata": {},
   "source": [
    "Let's use the identity $\\sum_{i=1}^n F_i^2 = F_n F_{n+1}$ to test `myfib`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "82689315-6f47-40d3-884b-c082383955d8",
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "for n = (10, 15, 20)\n",
    "    s = 0\n",
    "    for i = 1:n\n",
    "        s += (myfib(i))^2\n",
    "    end\n",
    "    @show s == myfib(n) * myfib(n+1)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "c7520625-3982-4887-bdd0-a678499f2744",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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