A091807 Given the infinite continued fraction i+(i/(i+(i/(i+...)))), where i is the square root of (-1), this is the denominator of the real part of the convergents.

1, 1, 2, 5, 3, 5, 41, 85, 178, 123, 769, 8, 3329, 533, 1602, 30005, 62441, 32485, 270409, 187575, 1171042, 2436961, 5071361, 26384, 1045821, 45703841, 95110562, 15225145, 411889609, 23809725, 1783745641, 3712008565, 7724760338

Offset

0,3

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=0..32.

Example

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

Mathematica

GenerateA091807[1] := I; GenerateA091807[n_] := I + I/(GenerateA091807[n-1]); GenerateDenominatorsA091807[n_] := Table[Denominator[Re[GenerateA091807[x]]], {x, 1, n}]; GenerateDenominatorsA091807[20] would give the first 20 terms.
A091807[n_] := Denominator[ Re[ Fold[ I/(I + #) &, 1, Range[n]]]]; Table[ A091807[n], {n, 0, 32}] (* Robert G. Wilson v, Mar 13 2004 *)

Crossrefs

Cf. A091806, A091808, A091809.

Sequence in context: A104978 A365723 A124568 * A085825 A198140 A339259

Adjacent sequences: A091804 A091805 A091806 * A091808 A091809 A091810

Keyword

cofr,frac,nonn

Author

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

Extensions

More terms from Robert G. Wilson v, Mar 13 2004

Status

approved