Science Buddies: "Ask an Expert"

I need to find the next two numbers in the following sequence.
0,13,13,21,34,55,86,139,216,342,550,898,?,?.

I know that the tens place is following a fibonacci sequence, but I can't figure out the pattern in the ones place. 0,13,13,21,34,55,86,139,216,342,550,898 = 0, 1, 1, 2, 5, 8, 13, 21, 34, 55, 89, 144, 233.

So I know what part of the answer would be for the missing two numbers, but I can't figure out the pattern in the ones place.
So the answer should be something like: 144(?) and 233(?). I'm not sure how to find that last number?

PLEASE HELP ASAP!

Hi Brad15,
I have to admit that I am stumped. Where are these numbers used in your experiment? Is this some kind of encryption key?
Any hints may help figure this out.
Please let us know.
Thanks!
Willey

Hi Brad15!
I'm not entirely sure how this could be solved, but I have noticed that for the first five/six terms, there are some interesting terms. When I saw that the numbers were all single digits that appeared to be random, my first instinct was to try modular arithmetic (a good introduction to modular arithmetic can be found here.)
When I added the previous two numbers, as in the Fibonacci sequence, and then put the resulting number in mod 5, I ended up with the next number:
0 + 3 mod 5 = 3
3 + 3 mod 5 = 6 mod 5 = 1
3 + 6 mod 5 = 9 mod 5 = 4
But this is where the pattern breaks down. My gut instinct is that this has something to do with the answer, but I'm not sure if it's related.
This is a very interesting problem! I'll keep working on it, and I agree with Willey that a hint on where these numbers are used would help greatly!

Thanks!
Vysarge

Hey guys, thanks for your help so far... My project is based on accomplishments of Fibonacci and my teacher created his own sequence that I have to solve.

Hi Brad15,

I'm still working on this and I seem to have hit a wall so far- I wanted to make sure you knew this was still being worked on! Do you know anything else about the sequence? Any information would help!

Thanks!
Vysarge

Brad15,

Here's a dumb idea, but it might work:

Write a small computer program that randomly generates sequences of instructions (or calls to subroutines within the program). The idea would be for it to generate hundreds of little programs that do the sorts of calculations needed to generate the sequence. The program would know the numbers that are expected, and it checks to see if the random program calculates them correctly. If so, you have a solution to your problem. If not, discard the little program and try the next one. You could limit it to addition, subtraction, multiplication, and division and have it create programs of varying length (you'll need to figure that out).

With a few hours work, if you have some programming experience, you can probably figure it out.


Good luck!