gistfile1.txt

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Evaluating the formula from:\n",
    "\n",
    "http://www.johndcook.com/blog/2015/08/20/when-is-a-triangle-a-square/"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import var, sqrt, init_printing\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAwAAAAJBAMAAAD0ltBnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARHa7IquJzRAy3VSZ\nZu8udG/tAAAARElEQVQIHWNgVHZRCmNgYGMtYuifwCDIZcDQL8DQAMT7GxhAOJyBgeE4A8NfDgaG\nZwy8P9gYWD4ysBnsZGB8wMBitAEAo3YOwwwd5UcAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$n$$"
      ],
      "text/plain": [
       "n"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "var(\"n\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAZgAAAArBAMAAAC3PIMJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\frac{1}{32} \\left(- 12 \\sqrt{2} + 17\\right)^{n} + \\frac{1}{32} \\left(12 \\sqrt{2} + 17\\right)^{n} - \\frac{1}{16}$$"
      ],
      "text/plain": [
       "                 n                  n     \n",
       "⎛       ___     ⎞    ⎛     ___     ⎞      \n",
       "⎝- 12⋅╲╱ 2  + 17⎠    ⎝12⋅╲╱ 2  + 17⎠    1 \n",
       "────────────────── + ──────────────── - ──\n",
       "        32                  32          16"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e = ((17+12*sqrt(2))**n + (17-12*sqrt(2))**n - 2 ) / 32\n",
    "e"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1\n",
      "2 36\n",
      "3 1225\n",
      "4 41616\n",
      "5 1413721\n",
      "6 48024900\n",
      "7 1631432881\n",
      "8 55420693056\n",
      "9 1882672131025\n",
      "10 63955431761796\n",
      "11 2172602007770041\n",
      "12 73804512832419600\n",
      "13 2507180834294496361\n",
      "14 85170343853180456676\n",
      "15 2893284510173841030625\n",
      "16 98286503002057414584576\n",
      "17 3338847817559778254844961\n",
      "18 113422539294030403250144100\n",
      "19 3853027488179473932250054441\n"
     ]
    }
   ],
   "source": [
    "for n_ in range(1, 20):\n",
    "    print n_, e.subs(n, n_).expand()"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}