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Search: id:A091809
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| A091809 |
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Given the infinite continued fraction i+(i/(i+(i/(i+...)))), where i is the square root of (-1), this is the denominator of the imaginary part of the convergents. |
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+0
7
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| 1, 1, 2, 5, 3, 10, 41, 85, 178, 123, 769, 10, 3329, 533, 1602, 30005, 62441, 64970, 270409, 187575, 1171042, 2436961, 5071361, 16490, 1045821, 45703841, 95110562, 15225145, 411889609, 47619450, 1783745641, 3712008565, 7724760338
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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The sequence of complex numbers (which this sequence is part of) converges to (i+sqrt(-1+4i))/2, found by simply solving the equation A=i+(i/A) for A using the quadratic formula. When plotted in the complex plane, these numbers form a counter-clockwise spiral that quickly converges to a point.
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EXAMPLE
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a(6)=10 since the sixth convergent is (3/5)+(13/10)i and hence the denominator of the imaginary part is 10.
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MATHEMATICA
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GenerateA091809[1] := I; GenerateA091809[n_] := I + I/(GenerateA091809[n-1]); GenerateDenominatorsA091809[n_] := Table[Denominator[Im[GenerateA091809[x]]], {x, 1, n}]; GenerateDenominatorsA091809[20] gives the first 20 terms.
A091809[n_] := Denominator[ IM[ Fold[ I/(I + #) &, 1, Range[n]]]]; Table[ A091809[n], {n, 0, 32}] (from Robert G. Wilson v Mar 13 2004)
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CROSSREFS
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Cf. A091806, A091807, A091808.
Sequence in context: A138765 A097753 A120860 this_sequence A110315 A094744 A129198
Adjacent sequences: A091806 A091807 A091808 this_sequence A091810 A091811 A091812
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KEYWORD
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cofr,frac,nonn
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AUTHOR
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Ryan Witko (witko(AT)nyu.edu), Mar 06 2004
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