79 lines
5.6 KiB
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79 lines
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<h2>Zeckendorf Representation</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 297</h3><span style="width:300px;color:#666;">Published on Friday, 18th June 2010, 05:00 pm; Solved by 1503</span></div>
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<p>Each new term in the Fibonacci sequence is generated by adding the previous two terms.<br />
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Starting with 1 and 2, the first 10 terms will be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.</p>
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<p>Every positive integer can be uniquely written as a sum of nonconsecutive terms of the Fibonacci sequence. For example, 100 = 3 + 8 + 89.<br />
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Such a sum is called the <b>Zeckendorf representation</b> of the number.</p>
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<p>For any integer <var>n</var>>0, let <var>z</var>(<var>n</var>) be the number of terms in the Zeckendorf representation of <var>n</var>.<br />
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Thus, <var>z</var>(5) = 1, <var>z</var>(14) = 2, <var>z</var>(100) = 3 etc.<br />
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Also, for 0<img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /><var>n</var><img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' />10<sup>6</sup>, ∑ <var>z</var>(<var>n</var>) = 7894453.</p>
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<p>Find ∑ <var>z</var>(<var>n</var>) for 0<img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' /><var>n</var><img src='images/symbol_lt.gif' width='10' height='10' alt='<' border='0' style='vertical-align:middle;' />10<sup>17</sup>.</p>
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