93 lines
6.2 KiB
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93 lines
6.2 KiB
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<h2>Rudin-Shapiro sequence</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 384</h3><span style="width:300px;color:#666;">Published on Sunday, 13th May 2012, 02:00 am; Solved by 241</span></div>
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<P>Define the sequence a(n) as the number of adjacent pairs of ones in the binary expansion of n (possibly overlapping).
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<BR/>E.g.: a(5) = a(101<sub>2</sub>) = 0, a(6) = a(110<sub>2</sub>) = 1, a(7) = a(111<sub>2</sub>) = 2</P>
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<P>Define the sequence b(n) = (-1)<sup>a(n)</sup>.
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<BR/>This sequence is called the <B>Rudin-Shapiro</B> sequence.</P>
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<P>Also consider the summatory sequence of b(n): <img src=project/images/p_384_formula.gif style="margin-top:-9px;">.</P>
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<P>The first couple of values of these sequences are:
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<BR/><TT>n   0   1   2   3   4   5   6   7
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<BR/>a(n)   0   0   0   1   0   0   1   2
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<BR/>b(n)   1   1   1   -1   1   1   -1   1
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<BR/>s(n)   1   2   3   2   3   4   3   4</TT></P>
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<P>The sequence s(n) has the remarkable property that all elements are positive and every positive integer k occurs exactly k times.</P>
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<P>Define g(t,c), with 1 <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> c <img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' /> t, as the index in s(n) for which t occurs for the c'th time in s(n).
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<BR/>E.g.: g(3,3) = 6, g(4,2) = 7 and g(54321,12345) = 1220847710.</P>
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<P>Let F(n) be the fibonacci sequence defined by:
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<BR/>F(0)=F(1)=1 and
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<BR/>F(n)=F(n-1)+F(n-2) for n>1.</P>
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<P>Define GF(t)=g(F(t),F(t-1)).</P>
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<P>Find ΣGF(t) for 2<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />t<img src='images/symbol_le.gif' width='10' height='12' alt='≤' border='0' style='vertical-align:middle;' />45.</P>
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