A091806 - OEIS

%I #22 Mar 15 2016 06:30:35

%S 0,1,1,3,2,3,26,53,111,77,480,5,2080,333,1001,18747,39014,20297,

%T 168954,117199,731679,1522639,3168640,16485,653440,28556241,59426081,

%U 9512831,257352966,14876567,1114503066,2319302053,4826511631,10044062391

%N 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.

%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.

%H Robert G. Wilson v, <a href="/A091806/b091806.txt">Table of n, a(n) for n = 1..1002</a>

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

%p A091806 := proc(n)

%p     numtheory[cfrac]([I,[I,I]$n-1]) ;

%p     numer(Re(%)) ;

%p end proc:

%p seq(A091806(n),n=1..100) ; # _Robert Israel_, Mar 14 2016

%t 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. *)

%t A091806[n_] := Numerator[ Re[ Fold[ I/(I + #) &, 1, Range[n]]]]; Table[ A091806[n], {n, 0, 32}] (* _Robert G. Wilson v_, Mar 13 2004 *)

%Y Cf. A091807, A091808, A091809.

%K cofr,frac,nonn

%O 1,4

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

%E More terms from _Robert G. Wilson v_, Mar 13 2004