A091808 Given the infinite continued fraction i+(i/(i+(i/(i+...)))), where i is the square root of (-1), this is the numerator of the imaginary part of the convergents.

1, 1, 3, 6, 4, 13, 53, 111, 231, 160, 1000, 13, 4329, 693, 2083, 39014, 81188, 84477, 351597, 243893, 1522639, 3168640, 6594000, 21441, 1359821, 59426081, 123666803, 19796382, 535556412, 61916837, 2319302053, 4826511631, 10044062391, 20901884640, 14499073000

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..35.

Example

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

Maple

A091808 := n -> numer(Im(numtheory[cfrac]([I, [I, I]$n-1]))):
seq(A091808(n), n=1..35); # Peter Luschny, Feb 27 2019

Mathematica

GenerateA091808[1] := I; GenerateA091808[n_] := I + I/(GenerateA091808[n-1]); GenerateNumeratorsA091808[n_] := Table[Numerator[Im[GenerateA091808[x]]], {x, 1, n}]; (* GenerateNumeratorsA091808[20] would give the first 20 terms. *)

Crossrefs

Cf. A091806, A091807, A091808, A091809.

Sequence in context: A292681 A299211 A067979 * A357235 A307460 A128719

Adjacent sequences: A091805 A091806 A091807 * A091809 A091810 A091811

Keyword

cofr,frac,nonn

Author

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

Extensions

More terms from Peter Luschny, Feb 27 2019

Status

approved