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%I #6 Dec 10 2015 02:37:44
%S 1,1,2,5,3,10,41,85,178,123,769,10,3329,533,1602,30005,62441,64970,
%T 270409,187575,1171042,2436961,5071361,16490,1045821,45703841,
%U 95110562,15225145,411889609,47619450,1783745641,3712008565,7724760338
%N 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.
%C 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 counterclockwise spiral that quickly converges to a point.
%e a(6) = 10 since the sixth convergent is (3/5) + (13/10)i and hence the denominator of the imaginary part is 10.
%t 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.
%t A091809[n_] := Denominator[ IM[ Fold[ I/(I + #) &, 1, Range[n]]]]; Table[ A091809[n], {n, 0, 32}] (* _Robert G. Wilson v_, Mar 13 2004 *)
%Y Cf. A091806, A091807, A091808.
%K cofr,frac,nonn
%O 1,3
%A Ryan Witko (witko(AT)nyu.edu), Mar 06 2004
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