{
"cells": [
{
"cell_type": "markdown",
"id": "161f1d9b-7001-4478-bbb8-ce1fd8043591",
"metadata": {},
"source": [
"##### Python for High School (Summer 2022)\n",
"\n",
"* [Table of Contents](PY4HS.ipynb)\n",
"* ![\"Open](\"https://colab.research.google.com/assets/colab-badge.svg\" "\\"Open")
\n",
"* [](https://nbviewer.org/github/4dsolutions/elite_school/blob/master/Py4HS_sphere_packing_classroom.ipynb)\n",
"\n",
"### Sphere Packing\n",
"\n",
"* [Generating the FCC](https://github.com/4dsolutions/Python5/blob/master/Generating%20the%20FCC.ipynb)\n",
"* [Crystal Ball Sequence](https://github.com/4dsolutions/Python5/blob/master/STEM%20Mathematics.ipynb)\n",
"* [OEIS A005901](https://oeis.org/A005901)\n",
"\n",
"\n",
"\n",
" 1 -- center ball\n",
"12 -- layer 1\n",
"42 -- layer 2\n",
"92 -- layer 3\n",
"...\n",
""
]
},
{
"cell_type": "code",
"execution_count": 41,
"id": "c081ad1b-0721-4369-ab5f-c56d2610813a",
"metadata": {},
"outputs": [],
"source": [
"def a005901(layer : int) -> int:\n",
" \"\"\"\n",
" https://oeis.org/A005901\n",
" layer >= 0\n",
" \"\"\"\n",
" if int(layer) < 0:\n",
" raise ValueError\n",
" if not (int(layer) == float(layer)):\n",
" raise ValueError\n",
" \n",
" if layer == 0:\n",
" return 1\n",
" return 10 * layer ** 2 + 2"
]
},
{
"cell_type": "code",
"execution_count": 37,
"id": "609a2d74-16a3-40d7-a177-026482891d39",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 12, 42, 92, 162, 252, 362, 492, 642, 812, 1002, 1212, 1442, 1692, 1962, 2252, 2562, 2892, 3242, 3612, 4002, 4412, 4842, 5292, 5762, 6252, 6762, 7292, 7842, 8412]\n"
]
}
],
"source": [
"print([a005901(x) for x in range(30)])"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "c33ed68d-9898-4be0-a18f-b7e06b576cff",
"metadata": {},
"outputs": [],
"source": [
"def accum_a005901(layer):\n",
" \"\"\"\n",
" A005902 cumulative version of A005901\n",
" \"\"\"\n",
" total = 0\n",
" for x in range(layer + 1):\n",
" total = total + a005901(x)\n",
" return total"
]
},
{
"cell_type": "markdown",
"id": "77cc6f74-56cf-4c42-b01a-dc8db611cd35",
"metadata": {},
"source": [
"Upon reading the notes for [A005902](https://oeis.org/A005902) we find there's a more direct way to compute the cumulative number of balls i.e. we do not have to sum the balls in each layer out to some target layer. Enter the target layer $L$ directly into this formula:\n",
"\n",
"$$\n",
"S = \\frac{(2L + 1)(5L^{2} + 5L + 3)}{3}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 43,
"id": "39f8127b-e480-4cde-8e42-0c678706fda7",
"metadata": {},
"outputs": [],
"source": [
"import sympy as sp"
]
},
{
"cell_type": "code",
"execution_count": 48,
"id": "e838d9d0-cd20-4037-a27a-2d41ab1986b3",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{\\left(2 L + 1\\right) \\left(5 L^{2} + 5 L + 3\\right)}{3}$"
],
"text/plain": [
"(2*L + 1)*(5*L**2 + 5*L + 3)/3"
]
},
"execution_count": 48,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = sp.Symbol('L')\n",
"expr = (2*L + 1)*(5*L**2 + 5*L + 3)/3\n",
"expr"
]
},
{
"cell_type": "code",
"execution_count": 55,
"id": "90e1cee6-54c0-4402-873f-1945efd1bc30",
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\displaystyle \\frac{10 L^{3}}{3} + 5 L^{2} + \\frac{11 L}{3} + 1$"
],
"text/plain": [
"10*L**3/3 + 5*L**2 + 11*L/3 + 1"
]
},
"execution_count": 55,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"formula = expr.expand()\n",
"formula"
]
},
{
"cell_type": "code",
"execution_count": 57,
"id": "a6464325-f5ac-4eae-a480-6f1f0de240ae",
"metadata": {},
"outputs": [],
"source": [
"def short_cut(layer):\n",
" if layer == 0:\n",
" return 1\n",
" return ((2 * layer + 1) * (5 * layer**2 + 5 * layer + 3)) // 3 # see below"
]
},
{
"cell_type": "code",
"execution_count": 61,
"id": "7fdd39c5-1013-4e7a-b759-45ba35d3522c",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3871"
]
},
"execution_count": 61,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"short_cut(10)"
]
},
{
"cell_type": "code",
"execution_count": 38,
"id": "50cf7bb2-9958-435c-ad6b-195af3467d07",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, 69355, 77197, 85609]\n"
]
}
],
"source": [
"print([short_cut(x) for x in range(30)])"
]
},
{
"cell_type": "code",
"execution_count": 63,
"id": "ef798a5c-acc1-440c-b034-73b8ddc7deee",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3871"
]
},
"execution_count": 63,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"int(formula.evalf(subs={L:10}))"
]
},
{
"cell_type": "code",
"execution_count": 56,
"id": "a24fecc2-5c61-4db1-9c5b-fd21210dbd0b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[1, 13, 55, 147, 309, 561, 923, 1415, 2057, 2869, 3871, 5083, 6525, 8217, 10179, 12431, 14993, 17885, 21127, 24739, 28741, 33153, 37995, 43287, 49049, 55301, 62063, 69355, 77197, 85609]\n"
]
}
],
"source": [
"print([int(expr.evalf(subs={L:x})) for x in range(30)])"
]
},
{
"cell_type": "markdown",
"id": "91958f11-5f72-4429-abcd-7dc81002597d",
"metadata": {},
"source": [
"### Remember how divsion works: / versus //"
]
},
{
"cell_type": "code",
"execution_count": 70,
"id": "7150b7c6-7664-4436-8079-6df7d54c9ebc",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"2.0"
]
},
"execution_count": 70,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"6 // 2.1"
]
},
{
"cell_type": "code",
"execution_count": 40,
"id": "3fe97b3e-617f-4a2b-baf1-a3462fabc53a",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"3"
]
},
"execution_count": 40,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"6 // 2"
]
},
{
"cell_type": "code",
"execution_count": 65,
"id": "690b4cbb-91e8-49f8-b19a-c182282dd2a7",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1"
]
},
"execution_count": 65,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"6 // 5"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "39fcc1fa-50f6-4afa-aba2-35f9b2d1382d",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3 (ipykernel)",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.12"
}
},
"nbformat": 4,
"nbformat_minor": 5
}