The other thing that comes to mind is that the cf is pretty unremarkable except for the one ginormous term, and that term is pretty much the same length as the big rational factor: 5193981023518027157495786850488117/7177905237579946589743592924684178 83364870763649235403921261388869364666045817819140268784224747492762 I'd bet that most of the information in that term comes from the rational multiplier itself. On Thu, Nov 12, 2015 at 11:30 AM, Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello funsters,
here is something which is a puzzle to me,
I was exploring numbers like Fibonacci(k)/(Lucas(k-n)*((1+sqrt(5))/2)^n, where n is small and k >> 1.
Here is the odd thing : if you expand for example the number
5193981023518027157495786850488117/7177905237579946589743592924684178/(1/2+1/2*5^(1/2))^2 into a continued fraction, the surprise comes from the partial quotients of that expansion. It is quite chaotic. The maximal value being 83364870763649235403921261388869364666045817819140268784224747492762,
What is this ? how come a simple number like a/b*sqrt(5) has a c.frac expansion which such values ? I thought that approximations of a number like sqrt(5) could not be like that.
A quick examination shows that the size of these numbers (the maximal value of the c.f expansion) will be like Fibonacci(k)^2 (if n is small). In this example we have,
n = 163 and n = 2.
Can someone tell me how is this possible ?
I really don't see a general formula, since for some values of n and k, the behavior of the c.f. is quite <normal> with no high values, what are the conditions to have the maximal value ? I made some programs to analyze this and found only bizarre examples.
Best regards,
ps : I am back on the math-fun list after a quick absence.
Simon Plouffe
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