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August 20, 2015 14:47
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{ | |
"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 | |
} |
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