A091809 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.

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

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..33.

Example

a(6) = 10 since the sixth convergent is (3/5) + (13/10)i and hence the denominator of the imaginary part is 10.

Mathematica

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}] (* Robert G. Wilson v, Mar 13 2004 *)

Crossrefs

Cf. A091806, A091807, A091808.

Sequence in context: A097753 A120860 A259971 * A110315 A275677 A221183

Adjacent sequences: A091806 A091807 A091808 * A091810 A091811 A091812

Keyword

cofr,frac,nonn

Author

Ryan Witko (witko(AT)nyu.edu), Mar 06 2004

Status

approved