101 lines
5.9 KiB
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101 lines
5.9 KiB
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<h2>Fibonacci Words</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 230</h3><span style="width:300px;color:#666;">Published on Saturday, 31st January 2009, 01:00 pm; Solved by 1738</span></div>
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<p>For any two strings of digits, A and B, we define F<sub>A,B</sub> to be the sequence (A,B,AB,BAB,ABBAB,...) in which each term is the concatenation of the previous two.</p>
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<p>Further, we define D<sub>A,B</sub>(<var>n</var>) to be the <var>n</var><sup>th</sup> digit in the first term of F<sub>A,B</sub> that contains at least <var>n</var> digits.</p>
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<p>Example:</p>
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<p>Let A=1415926535, B=8979323846. We wish to find D<sub>A,B</sub>(35), say.</p>
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<p>The first few terms of F<sub>A,B</sub> are:<br />
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1415926535<br />
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8979323846<br />
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14159265358979323846<br />
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897932384614159265358979323846<br />
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1415926535897932384689793238461415<span style="color: #FF0000"><b>9</b></span>265358979323846<br />
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</p>
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<p>Then D<sub>A,B</sub>(35) is the 35<sup>th</sup> digit in the fifth term, which is 9.</p>
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<p>Now we use for A the first 100 digits of π behind the decimal point:</p>
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<p>14159265358979323846264338327950288419716939937510 <br />
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58209749445923078164062862089986280348253421170679 </p>
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<p>and for B the next hundred digits:</p>
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<p>82148086513282306647093844609550582231725359408128 <br />
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48111745028410270193852110555964462294895493038196 .</p>
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<p>Find <img src='images/symbol_sum.gif' width='11' height='14' alt='∑' border='0' style='vertical-align:middle;' /><sub><var>n</var> = 0,1,...,17</sub> 10<sup><var>n</var></sup><img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' /> D<sub>A,B</sub>((127+19<var>n</var>)<img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />7<sup><var>n</var></sup>) .</p>
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