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Fibonacci sequence- advanced realations

  1. Feb 29, 2012 #1
    My main interest is usually physics, however I have become interested in the Fibonacci sequence. I looked at the sequence in detail and found a few interesting patterns. The main pattern was a relationship of the way that the digits of numbers compounded, the exact pattern was: 7,5,5,4,5,5... (that repeated). An easier way to explain it is that there are 7 single digit numbers, 5 double digit numbers, five triple digit numbers, 4 quadruple digit numbers etc. I was wondering if this was a mere coincidence (which is rare with Fibonacci numbers)or if it was relevant, and hopefully not pre-discovered (I thought I discovered Pascal's Triangle until i learned Blaise had beaten me by a couple hundred years...).
     
  2. jcsd
  3. Feb 29, 2012 #2
    Well the Pascal triangle was known well before Pascal was born.
     
  4. Feb 29, 2012 #3
    I strongly doubt there is a repeating pattern in that sequence you mention, which btw starts like this 7,5,5,4,5,5,5,4,5,5,5,5,4,5,5,5,5,4,5,5,5,4,5,5,5,5,4,..

    The sequence reflects the fact that the ratio between successive fibonaccis quickly approach the golden ratio [itex]\varphi[/itex]=(1+√5)/2, so that (log10[itex]\varphi[/itex])-1= 4.784.. becomes the average number in the sequence. Any repeating pattern depends on that last number being rational, which, uhm.. lacking something better to say, is unlikely.
     
    Last edited: Feb 29, 2012
  5. Mar 4, 2012 #4
    Re: Fibonacci sequence- advanced relations

    In general, Rocketboy123,

    You might want to familiarize yourself with the properties of Lucas Sequences (not "Lucas Numbers"). It will save you many headaches later on.

    - AC
     
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