The Condensed Number Value Pattern in the Fibonacci Series

The Fibonacci numbers (FN) are a sequence of integers, starting with 0, 1 and continuing 1, 2, 3, 5, 8, 13, ..., each new number being the sum of the previous two. The Fibonacci numbers, and in conjunction the golden ratio, are a popular theme in culture. They have been mentioned in novels, films, television shows, and songs. The numbers have also been used in the creation of music, visual art, and architecture.

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657

A condensed number value (CNV) is derived by repeatedly summing the digits of an integer until a single digit remains.

Examples:

12 => 1 + 2  = _CNV3
66 => 6 + 6 = 12 by previous example condenses to _CNV3
So 12 and 66 have the same condensed value of _CNV3

Condensed Number Value of the Fibonacci series, first 28 values:

0 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9 1 1 2


0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657

10946 => 1+0+9+4+6 = 20, then 2+0 => _CNV2
17711 => 1+7+7+1+1 = 17, then 1+7 => _CNV8
28657 => 2+8+6+5+7 = 28, then 2+8 = 10, then 1+0 => _CNV1
46368 => 4+6+3+6+8 = 27, then 2+7 => _CNV9
75025 => 7+5+0+2+5 = 19, then 1+9 = 10, then 1+0 => _CNV1
121393 => 1+2+1+3+9+3 = 19, then 1+9 = 10, then 1+0 => _CNV1
196418 => 1+9+6+4+1+8 = 29, then 2+9 = 11, then 1+1=> _CNV2

My recent discovery is that the sequence of Fibonacci numbers follow a repeating pattern of CNV. I searched for a repeating pattern, and found it at "_CNV1 _CNV1" in the 26th and 27th positions, 75025 and 121393, which are the same as the 2nd and 3rd positions.

Fibonacci numbers are derived by adding. Condensed Number Values for the summation process of integers follows a table:

Integer + Integer = CNV ...and... CNV + CNV = CNV

X + Y = Z where X and Y are either Integers or CNV's, and Z is a CNV

1 + 1 = 2
1 + 2 = 3
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
1 + 7 = 8
1 + 8 = 9
1 + 9 = 1
2 + 2 = 4 .... (and 2 + 1 = 3, so I didn't repeat Y + X = Z when I have already stated X + Y = Z)
2 + 3 = 5
2 + 4 = 6
2 + 5 = 7
2 + 6 = 8
2 + 7 = 9
2 + 8 = 1
2 + 9 = 2
3 + 3 = 6
3 + 4 = 7
3 + 5 = 8
3 + 6 = 9
3 + 7 = 1
3 + 8 = 2
3 + 9 = 3
4 + 4 = 8
4 + 5 = 9
4 + 6 = 1
4 + 7 = 2
4 + 8 = 3
4 + 9 = 4
5 + 5 = 1
5 + 6 = 2
5 + 7 = 3
5 + 8 = 4
5 + 9 = 5
6 + 6 = 3
6 + 7 = 4
6 + 8 = 5
6 + 9 = 6
7 + 7 = 5
7 + 8 = 6
7 + 9 = 7
8 + 8 = 7
8 + 9 = 8
9 + 9 = 9

So after a pattern of  24 Condensed Number Values:

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

the Fibonacci series starts repeating using this pattern. Some interesting features of this distribution of CNV:

 a) There are five _CNV1's and five _CNV8's
 b) There are two of the other values _CNV2,_CNV3,_CNV4,_CNV5,_CNV6,_CNV7,_CNV9

I noticed that Fibonacci numbers have a distribution of 2 odd numbers for each even value, a 2:1 ratio, with the pattern after the first zero value being odd, odd, even, odd, odd, even, .... repeating forever, because two odd numbers sum to an even, and then an even and odd produce an odd.

Because the smaller repeating group of 3 (odd, odd, even) is a factor of 24, they group together on an infinitely repeating basis:

CNV: 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

FN:  O O E O O E O O E O O E O O E O O E O O E O O E <-Odd Even

If FN has a _CNV2, then FN is even

If FN has a _CNV3, then FN is odd 

If FN has a _CNV4, then FN is odd 

If FN has a _CNV5, then FN is odd  

If FN has a _CNV6, then FN is odd 

If FN has a _CNV7, then FN is even 

If FN has a _CNV9, then FN is even 

If FN has a _CNV1, then FN is even 1/5 of the time, odd 4/5

If FN has a _CNV8, then FN is even 1/5 of the time, odd 4/5  

^Note: I have had a lot of fun researching the properties of Condensed Number Values since about age 10. When I was a child in church services (and bored), I would find the CNV of the offering and attendance numbers that were posted on the wall in the church sanctuary.

Please let me know if anything is unclear or if you believe I have made a mistake in my math, logic, or presentation. Thank you!

 

Update: 2012.09.08 - I found another website where someone else had published some of the same things I mentioned in this blog post. It has a publication date of May 2012, but I didn't copy that work.

Link: Fibonacci 24 Pattern