A091806 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.

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

Offset

1,4

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

Robert G. Wilson v, Table of n, a(n) for n = 1..1002

Example

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

Maple

A091806 := proc(n)
    numtheory[cfrac]([I, [I, I]$n-1]) ;
    numer(Re(%)) ;
end proc:
seq(A091806(n), n=1..100) ; # Robert Israel, Mar 14 2016

Mathematica

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

Crossrefs

Cf. A091807, A091808, A091809.

Sequence in context: A350517 A123170 A253381 * A248244 A139170 A139075

Adjacent sequences: A091803 A091804 A091805 * A091807 A091808 A091809

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