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I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it to powers 1..n then rounding that number to generate a sequence of Lucas numbers.

I wondered what relationship (difference) the actual number had with the rounded number so I wrote some Python 3 code (I tried it in Octave also):

phi = (1 + math.sqrt(5)) / 2.0
for n in range(2000):
   number = math.pow(phi, n)
   print(n, round(number) - number)

It looks like after $\phi^{73}$ the value drops to zero (well, Python and Octave's 0.0) and never recovers on my machine. $\phi^{1259}$ produces an overflow. I just thought this was an interesting property and just was wondering:

  1. Are the power values after 73 really zero or just too small for my 64-bit machine?
  2. Is the sequence shown here used or applied anywhere? I searched in some of the Lucas literature but nothing popped out about the differences.
  3. Does anyone know if there is any more variance after 73?

Plot of values up to 73

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  • 2
    $\begingroup$ It's just double precision arithmetic. Generally, you have $L_n = \phi^n + \psi^n$, where $\psi = 1-\phi = \frac{1-\sqrt{5}}{2}$. $\endgroup$ – Daniel Fischer Oct 6 '14 at 12:33

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