Fibonacci sequence- advanced realations

Rocketboy123

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...).

epsi00

Well the Pascal triangle was known well before Pascal was born.

Norwegian

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.

Anti-Crackpot

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