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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.
7
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.
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
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