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I am away for a little while working on a few or more episodes for The Adventures of My Space Alien Alter Ego story ... will return (to Earth) soon!

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esbb

2012-08-29

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.

Rob Z Tobor said...

The odd odd even seemed odd, it sort of scrambled my brain a bit.

I might just stick with Mr Fibonacci's famous Italian soft ice cream.

What does Mrs ESB think, I am sure she must agree with you....

esbboston said...

Rob: As you might imagine, being a good spouse, she always agrees with me. (hahahahahahahahahahahaaaahhhahahaaa)
It sort of scram-bled my brain a little bit too and too-k a long time to write. I enjoy math, puzzles and numbers.

CNV also work for multiplication. Here are a couple examples.

2 x 2 = 4
11 x 11 = 121, so 1+2+1=CNV of 4
29 x 101 = 2929, so 2+9+2+9=22, so 2+2 = CNV of 4

4 x 4 = 16, so 1+6=7
13x1111=14443, so 1+4+4+4+3=16, so 1+6= CNV of 7

There is a principle that 9's drop out, so sometimes it is very easy to calculate the CNV of large numbers by ignoring nines and groups of other numbers that equal nine. Example: 49999999 has a CNV of 4

4+9+9+9+9+9+9+9=67, so 6+7=13, so 1+3= CNV of 4

So building on the previous multiplication that 4 x 4 = CNV of 7, then the product of 49999999x49999999 has a CNV of 7.

Fun with numb-bers. The b in numb looks like an upside down rotated nine, so maybe that eXplains why it is silent?

Rob Z Tobor said...

I have a feeling I may not like the answer to this question but sometimes a man has a ask a question that a man should not ask.

So what happens if you add fractions.

esbboston said...

Rob: Simple answer, CNV is a mathematics of integers.

Aysh said...

I have concluded from this post, my good sir, that you are a very clever man.
My brain got confused by all the numbersssersers

esbboston said...

Aysh: Can you teLL that I get great joy from eXploring aLL this stuff? I am not sure about the original-ness of my discovery, I am sure there are bound to be other people who have noticed these things, as famous as Fibonacci is to math folks. I semi-constantly look for patterns, order and symmetry in things, they bring me comfort.

Pearl said...

Very cool!

Have you seen "Pi"? I think you would like it.

http://en.wikipedia.org/wiki/Pi_(film)

Pearl

esbboston said...

Pearl: I tried to watch that movie, but just could not finish it. At the time I was playing with pi out to 200 places, so when they blatantly had an incorrect sequence of numbers for pi on the opening credits, it bothered me too much. I had it on a DVR at that time so I was able to slow it down to clearly see their mistake. They claimed to have made that movie for only \$60,000. Amazing, with box office receipts in the three million range? AND ... he made that movie in an eXpensive place like New York City ... just think how cheap he could have made it in Fargo, North Dakota, or better yet, Brainerd.

Marianne said...

I am so dumb.

esbboston said...

Marianne: What?!?!? You be not dumb!!! The neXt time I go to New York City, I can spend some time with you eXplaining Condensed Numbers in a more organized approach. Perhaps I can get a textbook published, now that I have this little chapter on Fibonacci. I had several pages put together when I was thinking about math grad school about 15 years ago. I realize that you don't live in New York City, but it would give us both a good eXcuse to go to NYC. I could teLL my wife, "Hey, lets go to NYC to a Number Theory Convention so I can teach a seminar!" and you could teLL your family, "FoM, lets aLL go to NYC to a Number Theory Convention, and continue our eXploration of useless from a daily perspective mathematics!" (FoM=Family of Marianne) Have you ever been to the Metropolitan Museum of Art? It is fantastic. I have been there twice in real life, I think 1994 & 1995. Everytime I go to NYC in Google Earth I try to remember to stop by the museum.

Rob Z Tobor said...

I have been to NY on Google Earth too.

I'm sure there is an old saying which goes something like

The sum of the whole is greater than the parts of the sum.

or was it Divide and conquer.

esbboston said...

Rob: I was just at NYC in GoogleEarth, but I did not see you. I am anxiously waiting for Google Mars.

I wiLL go with the saying "The whole summer is great." so that could mean someone who performs addition on integers is outstanding, or the weather in the warmest season is wonderful.

I have had a difficult day of automotive repair. It took me too long to find the almost right tool and the almost wrong tool in order to change the second fuel filter on my truck. I needed a 34 mm socket but the only one in town wound up being too long, but the shorter 35 mm wound up working, but the space was so confined that the ratchet mechanism just barely worked. I think I wiLL try to get a universal joint in a 1/2 inch drive socket before neXt time.

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