I can't spell Fibinacci

[bumab]

I've got a pattern question:

 

For the sequence 1 2 3 6 11 20 37 68 125 230 423 778 ...

 

two patterns are apparent.

 

1) the fibinacci (sp?) pattern, each number is the sum of the previous 3 numbers

2) each number is the previous number doubled, minus the number 3 places before the previous number

i.e. for 37= 20 x 2 - 3

for 68 = 37 x 2 - 6

 

Can you show that those two solutions are agabrically equivalent? That 2nd solution works for other number patterns where each number is the sum of the previous 3, whether the origin is 1 or not, like 3 4 7 14 25 46 ...

[Tormod]

The Fibonacci sequence is supposedly only two numbers, not three...as in 1 1 2 3 5 8 13 21 34 etc...

[bumab]

Hmmmm.... I just talked to a math teacher, he claims 3.

 

But it doesn't really matter for the pattern... how would one go about looking for agabraic equivalence?

[bumab]

Well, looks like you're right about it only being two numbers. So nevermind my now poorly titled origional question.

 

Although the same result applies-

 

1 1 2 3 5 8 13 21 33

 

3 x 2 -1 = 5

5 x 2 -2 = 8

8 x 2 -3 = 13

13 x 2 - 5 = 21

 

The real question I have: How would one go about looking for algabraic equivalence between those two solutions to that pattern?

[Turtle]

___Well...I feel duty bound to answer & my answer is...I don't know. If you tell us which type of math class, it may lend a clue as to the type of solution sought. It might have a solution in matrices or other linear programming method or just as you say strictly algebraic. I simply don't know how to set it up.

___If you are studying Fibonacci Numbers, look up 'phylotaxy'; very interesting. ;)

[bumab]

Not a math class at all. I'm a biology teacher, and I was playing with the math clubs problem sets. There was a bunch of pattern recognition problems... I was just curious if those two solutions were identical or not.

[C1ay]

Fibonacci numbers...

[bumab]

Interesting... thanks!

[Turtle]

___Their explanation of 'phylotaxis' is a bit vauge. To clarify, here is how you establish the phylotaxy of a specific plant. Find a branch which is a single year's growth. Beginning with the bottom leaf (the oldest) move up the stem section until arriving at a leaf which is directly above the starting one. Now count how many leaves lie between; this is the denominator. Now, count how many times the arrangment of said leaves wind around the stem before you reach the top leaf as above; this number is the numerator.

___These numbers, regardless of species, are always Fibonacci Numbers.

___I have always wondered exactly how DNA codes mathematics such as this.

;)

[Tormod]

___I have always wondered exactly how DNA codes mathematics such as this.

;)

 

Most likely it is the optimal distribution of branches to get the most sunlight.

[Turtle]

___But that doesn't explain all the variations on the theme. The phylotaxy of the Oak is not the phylotaxy of the Maple, is not that of the Cypress, etc, yet they all 'successfully' use the same sunlight.

___I more had in mind some idea like since we have the 4 chemicals that form all DNA, that maybe this is equivalent to a base 5 math system, & that somewhere in a gene sequence you could find the equivalent of Fibonacci Numbers (or their expression/formula) expressed in base 5. Well, you know me & math; have to earn my title. ;)

[Qfwfq]

The real question I have: How would one go about looking for algabraic equivalence between those two solutions to that pattern?

It isn't difficult to show the equivalence. Start from the recursive rule:

 

1) f(n + 2) = f(n) + f(n + 1),

 

write it for the incremented n:

 

2) f(n + 3) = f(n + 1) + f(n + 2),

 

and show this is equivalent to the other recursive rule:

 

3) f(n + 3) = 2f(n + 2) - f(n)

 

by making use of formula (1) to replace the terms f(n + 2) in both (2) and (3). The equivalence follows quite easily.