72 lines
5.7 KiB
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72 lines
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<h2>Polynomials of Fibonacci numbers</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 435</h3><span style="width:300px;color:#666;">Published on Saturday, 7th September 2013, 04:00 pm; Solved by 97</span></div>
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<p>The <strong>Fibonacci numbers</strong> {f<sub>n</sub>, n <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 0} are defined recursively as f<sub>n</sub> = f<sub>n-1</sub> + f<sub>n-2</sub> with base cases f<sub>0</sub> = 0 and f<sub>1</sub> = 1.</p>
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<p>Define the polynomials {F<sub>n</sub>, n <img src='images/symbol_ge.gif' width='10' height='12' alt='≥' border='0' style='vertical-align:middle;' /> 0} as F<sub>n</sub>(x) = <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' />f<sub>i</sub>x<sup>i</sup> for 0 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> i <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> n.</p>
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<p>For example, F<sub>7</sub>(x) = x + x<sup>2</sup> + 2x<sup>3</sup> + 3x<sup>4</sup> + 5x<sup>5</sup> + 8x<sup>6</sup> + 13x<sup>7</sup>, and F<sub>7</sub>(11) = 268357683.</p>
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<p>Let n = 10<sup>15</sup>. Find the sum [<img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /><sub>0<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />x<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />100</sub> F<sub>n</sub>(x)] mod 1307674368000 (= 15!).</p>
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