Pattern of sum of digits of fibonacci numbers - Part I

I think Fibonacci is one of the most studied sequences in mathematics and which is named after Leonardo Fibonacci. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture (from Wikipedia).

It is used in many real-world applications like stock markets predictions (e.g. retracement, etc). I was also fascinated so I started looking at pattern formed by Fibonacci numbers and I found out few amazing things about it:-

Java code for printing → Fibonacci Series (left/right/normal aligned) → FiboSeries.java

Important → 
Why am I doing it? 
1. for the love of numbers and maths, and 
2. I wanted to find (which I haven't found yet) n-th Fibonacci number without going through all previous Fibonacci numbers, 
or at least find out how many digits are there in any n-th Fibonacci number 
or what is the sum of all digits of such n-th Fibonacci number.

1. Fibonacci numbers having same number of digits (i.e. consecutive Fibonacci numbers having same number of digits) will be either 4 or 5. 

(The only exception being the first two numbers i.e. 0 and 1 - which are one digit numbers, along with next 5 numbers → 1, 2, 3, 5, 8, making total 7 1-digit numbers in Fibonacci series).

e.g. There are four Fibonacci numbers having 4-digits, they are  →  1597, 2584, 4181, 6765
and there are five 5-digit Fibonacci numbers, they are → 10946, 17711, 28657, 46368, 75025, and so on.

Here are few numbers which I manually wrote (at first and then I wrote a program to print them all)

Fig 1 - First few Fibonacci numbers by number of digits group

IMP - "But when it will have 5 numbers and when it will have 4 is a pattern in itself." 
So lets find out (bare with me for next few pictures, its kind of easy but was difficult write what are those pictures and what are those numbers in following pictures)...

In order to find this, I wrote down around 500 first Fibonacci numbers and their digit's sum.
Following are the photos of those pages on which I wrote those digit's sum.

How to read these next pictures of pages → I just consider ( or use) sum of digits of Fibonacci numbers and in the column next to them is number of digits in those 4 or 5 consecutive Fibonacci numbers.

e.g. take the first few lines from next picture are →
1    2    3    5    8    -    1
4    3    7    1    8    -    2
9    8    8    7    6    -    3
4    1    5    6          -    4

2    8    1    9    1    -    5
1    2    3    5    8    -    6
4    3    7    1    8    -    7
9    8    8    7          -    8

here is how to read them →
1    2    3    5    8    -    1 (i.e. 1-digit Fibonacci numbers)   
(five 1-digit numbers i.e. 1 (Sum of Digits = 1), 2 (Sum of Digits = 2), 3 (Sum of Digits = 3), 5 (Sum of Digits = 5), 8 (Sum of Digits = 8))

4    3    7    1    8    -    2 (i.e. 2-digit Fibonacci numbers)
(five 2-digit Fibonacci Numbers, i.e. 13 (Sum of Digits = 4), 21 (Sum of Digits = 3), 34 (Sum of Digits = 7), 55 (Sum of Digits = 1), 89 (Sum of Digits = 8)) 

9    8    8    7    6    -    3 (i.e. 3-digit Fibonacci numbers)
(five 3-digit Fibonacci Numbers i.e. 144 (Sum of Digits = 9), 233 (Sum of Digits = 8), 377 (Sum of Digits = 8), 610 (Sum of Digits = 7), 987 (Sum of Digits = 6)) 

4    1    5    6          -    4 (i.e. 4-digit Fibonacci numbers)   
(four 4-digit Fibonacci Numbers i.e. 1597 (Sum of Digits = 4), 2584 (Sum of Digits = 1), 4181 (Sum of Digits = 5), 6765 (Sum of Digits = 6))

2    8    1    9    1    -    5  (i.e. 5-digit Fibonacci numbers)
(five 5-digit Fibonacci Numbers i.e. 10946 (Sum of Digits = 2), 17711 (Sum of Digits = 8), 28657 (Sum of Digits = 1), 46368 (Sum of Digits = 9), 75025 (Sum of Digits = 1))

1    2    3    5    8    -    6 (i.e. 6-digit Fibonacci numbers)
(five 6-digit Fibonacci Numbers i.e. 121393 (Sum of Digits = 1), 196418 (Sum of Digits = 2), 317811 (Sum of Digits = 3), 514229 (Sum of Digits = 5), 832040 (Sum of Digits = 8))

4    3    7    1    8    -    7 (i.e. 7-digit Fibonacci numbers)
(five 7-digit Fibonacci Numbers i.e. 1346269  (Sum of Digits = 4), 2178309  (Sum of Digits = 3), 3524578  (Sum of Digits = 7), 5702887  (Sum of Digits = 1), 9227465 (Sum of Digits = 8))

9    8    8    7          -    8 (i.e. 8-digit Fibonacci numbers)
(four 8-digit Fibonacci Numbers i.e. 14930352 (Sum of Digits = 9), 24157817 (Sum of Digits = 8), 39088169 (Sum of Digits = 8), 63245986 (Sum of Digits = 7))

and so on... 

Lets start all over again from 1-digit Fibonacci till very large Fibonacci numbers of say, 431 digits →

Here for each group, I have written either 4 or 3 on left (over curly braces), those numbers mean that there are 3 or 4 groups of 5 same-digit-length-n-consecutive Fibonacci Number groups


IMP - these numbers will give us the pattern when does a group of 5 same-digit-length-n-consecutive Fibonacci numbers occur and when a group of 4 same-digit-length-n-consecutive Fibonacci numbers are going to occur?. 

Fibonacci number's digit's sum from 1-digit till 45-digit Fibonacci numbers

Fibonacci number's digit's sum from 46-digit till 111-digit Fibonacci numbers

Fibonacci number's digit's sum from 112-digit till 179-digit Fibonacci numbers

Fibonacci number's digit's sum from 180-digit till 249-digit Fibonacci numbers

Fibonacci number's digit's sum from 250-digit till 320-digit Fibonacci numbers

Fibonacci number's digit's sum from 321-digit till 390-digit Fibonacci numbers

Fibonacci number's digit's sum from 391-digit till 431-digit Fibonacci numbers

From all above calculations, if we take numbers those which are written over curly braces and try to find a pattern that how many times 5 same-digit-length-n-consecutive Fibonacci numbers and when does 4 same-digit-length-n-consecutive Fibonacci numbers occur.

The pattern appears to be - 
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,   
(meaning 4, 4 pair appears 6 times interleaved with 3s)
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  
(meaning 4, 4 pair appears 7 times interleaved with 3s)
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  
(meaning 4, 4 pair appears 6 times interleaved with 3s)
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4, 
(meaning 4, 4 pair appears 7 times interleaved with 3s)
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  
(meaning 4, 4 pair appears 7 times interleaved with 3s)
3, 4, 3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  3, 4,  4,  
(meaning 4, 4 pair appears 6 times interleaved with 3s)

and so on...

Interpretation  (it was very complex to come up with this - please ignore if you don't get it easily ;))  →
"Occurrence of 5 same-digit-length-n-consecutive Fibonacci numbers 4 times twice (i.e. 4, 4) interleaved with 5 same-digit-length-n-consecutive Fibonacci numbers 3 times once before hitting next occurrence of 5 same-digit-length-n-consecutive Fibonacci numbers 4 times only once vibrates between 6, 7, 6, 7....

Note - All 3 times and 4 times occurrences are interleaved with 4 same-digit-length-n-consecutive Fibonacci numbers once"

meaning there is a pattern inside a pattern, which is  →

6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7 so on...

Look at the circled 4 (single 4) in picture below, I have written it.

Repeating pattern in Fibonacci number's digit's sum

I don't know if someone can benefit from all this, but its kind of interesting that there are so many patterns in Fibonacci series numbers. I think numbers are just awesome. Even Prime numbers seems to have weirdo patterns.

Note: Watch Prime Spirals → a video by Numberphile  → https://www.youtube.com/watch?v=iFuR97YcSLM

2. Sum of digits of Fibonacci numbers also form a Fibonacci series and this series is → "1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 9, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 9, 1"
Above series is a 24 numbers long and it circles around once it reaches its ends. 
(Apparently this is already known fact about Fibonacci numbers, please refer 
→ http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html)

Important Links →
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
http://en.wikipedia.org/wiki/Fibonacci_number
http://en.wikipedia.org/wiki/Golden_ratio

Important Edit →

While writing this whole thing, I didn't know that there already exists a formula to calculate any n-th Fibonacci number using golden ratio (pi = 1.61803398874989484820...).

And yes Fibonacci numbers are whole numbers so take nearest whole number of above equations solution for given n.

Which solves above problem. Here is that link →
http://www.mathsisfun.com/numbers/fibonacci-sequence.html
http://www.mathsisfun.com/numbers/golden-ratio.html