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Search: id:A091806
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| A091806 |
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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 numerator of the real part of the convergents. |
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+0
7
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| 0, 1, 1, 3, 2, 3, 26, 53, 111, 77, 480, 5, 2080, 333, 1001, 18747, 39014, 20297, 168954, 117199, 731679, 1522639, 3168640, 16485, 653440, 28556241, 59426081, 9512831, 257352966, 14876567, 1114503066, 2319302053, 4826511631, 10044062391
(list; graph; listen)
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OFFSET
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1,4
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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)=3 since the sixth convergent is (3/5)+(13/10)i and hence the numerator of the real part is 3.
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MATHEMATICA
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GenerateA091806[1] := I; GenerateA091806[n_] := I + I/(GenerateA091806[n-1]); GenerateNumeratorsA091806[n_] := Table[Numerator[Re[GenerateA091806[x]]], {x, 1, n}]; GenerateNumeratorsA091806[20] would give the first 20 terms.
A091806[n_] := Numerator[ Re[ Fold[ I/(I + #) &, 1, Range[n]]]]; Table[ A091806[n], {n, 0, 32}] (from Robert G. Wilson v Mar 13 2004)
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CROSSREFS
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Cf. A091807, A091808, A091809.
Sequence in context: A092950 A059239 A123170 this_sequence A139170 A139075 A089750
Adjacent sequences: A091803 A091804 A091805 this_sequence A091807 A091808 A091809
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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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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Mar 13 2004
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