Fibonacci numbers


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Profile Paul D Harris
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Message 1207762 - Posted: 19 Mar 2012, 7:46:09 UTC
Last modified: 19 Mar 2012, 7:46:34 UTC

Is there an app that will run on my computer that will calculate Fibonacci numbers similar to a Pi program?
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Message 1207781 - Posted: 19 Mar 2012, 10:33:25 UTC - in response to Message 1207762.
Last modified: 19 Mar 2012, 10:33:51 UTC

I don't know but Enrico Bombieri of the Princeton Institute for Advanced Study mentions a "Fibonacci quarterly" in an article on Fibonacci included in a book "Fibonacci tra arte e scienza" published by Cassa Di Risparmio di Pisa with Luigi A.Radicati di Brozolo (ed). They should know.
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Message 1207833 - Posted: 19 Mar 2012, 14:35:38 UTC
Last modified: 19 Mar 2012, 14:36:09 UTC

I think Fibonacci sequences are also involved in some Fractals, particularly the Mandelbrot Set.

see http://www.sunflowerblog.ch/2007/06/03/the-fibonacci-numbers-and-mandelbrots-fractals/.


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Message 1207865 - Posted: 19 Mar 2012, 16:21:16 UTC - in response to Message 1207762.
Last modified: 19 Mar 2012, 16:21:36 UTC

Is there an app that will run on my computer that will calculate Fibonacci numbers similar to a Pi program?

Sure. It's pretty easy to make even on your own.

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Message 1207880 - Posted: 19 Mar 2012, 16:42:21 UTC - in response to Message 1207833.

I think Fibonacci sequences are also involved in some Fractals, particularly the Mandelbrot Set.


Yeah they pop up all over in mathematics. Phi (the golden ratio) comes immediately to mind.
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Message 1207911 - Posted: 19 Mar 2012, 17:21:10 UTC

Another number, a ratio kept popping into my mind as I wrote the previous post that featured the golden ratio of 1.618. The number was 1.4. It turns out that there is another “golden” ration based on 1.414, which is the square root of 2, which also happens to be the length of the diagonal of a square with sides of length 1.

Profile William Rothamel
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Message 1207930 - Posted: 19 Mar 2012, 17:54:59 UTC - in response to Message 1207911.
Last modified: 19 Mar 2012, 17:57:40 UTC

There is a "Binet" formula and also a generating function where you round to get the answer. Do a Google and you can get to other generating sites as well. If you are interested in Fibonacci numbers, the golden ration and hidden structure in Mathematics, contact me via private message and I will send you a paper that I wrote a few years back on the subject.

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Message 1207989 - Posted: 19 Mar 2012, 19:16:02 UTC

If you average the ratios of adjacent Fibonacci numbers: 1,1,2,3,5,8,13,21... with Lucas number ratios: 2,1,3,4.7,11,18,29... you can approach Phi, to any desired level of accuracy, in fewer steps than using either alone. Michael

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Message 1208115 - Posted: 20 Mar 2012, 2:19:55 UTC

(1 + sqrt(5)) / 2

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Message 1208119 - Posted: 20 Mar 2012, 2:23:13 UTC

a:=1;
> b:=1;
>
> for k from 1 to 100 do
> c:=a+b:
> print(c);
> a:=b:
> b:=c:
> od:

a := 1


b := 1


2


3


5


8


13


21


34


55


89


144


233


377


610


987


1597


2584


4181


6765


10946


17711


28657


46368


75025


121393


196418


317811


514229


832040


1346269


2178309


3524578


5702887


9227465


14930352


24157817


39088169


63245986


102334155


165580141


267914296


433494437


701408733


1134903170


1836311903


2971215073


4807526976


7778742049


12586269025


20365011074


32951280099


53316291173


86267571272


139583862445


225851433717


365435296162


591286729879


956722026041


1548008755920


2504730781961


4052739537881


6557470319842


10610209857723


17167680177565


27777890035288


44945570212853


72723460248141


117669030460994


190392490709135


308061521170129


498454011879264


806515533049393


1304969544928657


2111485077978050


3416454622906707


5527939700884757


8944394323791464


14472334024676221


23416728348467685


37889062373143906


61305790721611591


99194853094755497


160500643816367088


259695496911122585


420196140727489673


679891637638612258


1100087778366101931


1779979416004714189


2880067194370816120


4660046610375530309


7540113804746346429


12200160415121876738


19740274219868223167


31940434634990099905


51680708854858323072


83621143489848422977


135301852344706746049


218922995834555169026


354224848179261915075


573147844013817084101


927372692193078999176

>

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Message 1208120 - Posted: 20 Mar 2012, 2:25:08 UTC

a:=1;
>
> for k from 1 to 10 do
> b:=a*(1+sqrt(5))/2:
> print(evalf(b));
> a:=b:
> od:

a := 1


1.618033989


2.618033990


4.236067980


6.854101971


11.09016995


17.94427193


29.03444188


46.97871382


76.01315574


122.9918696

>

Profile William Rothamel
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Message 1208125 - Posted: 20 Mar 2012, 2:47:10 UTC - in response to Message 1207989.
Last modified: 20 Mar 2012, 2:50:00 UTC

If you average the ratios of adjacent Fibonacci numbers: 1,1,2,3,5,8,13,21... with ===


actually you can start with any numbers--say two arbitrary three digit numbers. Add them together and then add this sum to the 2nd number and then continue on by adding this sum to the previous one and so on. You will soon approach PHI at the tenth number if you take the ratio of the tenth to the ninth. This ratio will be at least 1.61 and possibly you could round it to 1.618. Try it for yourself--makes a good magic trick.

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Message 1208175 - Posted: 20 Mar 2012, 6:10:01 UTC
Last modified: 20 Mar 2012, 6:23:05 UTC

Nice respondence to Fibonacci numbers thanks.
I am not a math wize I took my math in the old days before computers and we use slide rules so what does a:=1 mean
As a part of the symbol sometimes used to mean "A is defined as 1" or is a=1?
And what does === mean?
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Message 1208220 - Posted: 20 Mar 2012, 12:08:26 UTC

the : means a variable in computer languages.

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Message 1208227 - Posted: 20 Mar 2012, 12:39:22 UTC - in response to Message 1207762.

Is there an app that will run on my computer that will calculate Fibonacci numbers similar to a Pi program?


They're 'organic' ratios, rather than absolute numbers. just use the golden ratio ;)

See any Fibonacci expansion contours or Golden Ratios in the speakers I made when I was 17?

and here's some fibonacci sequences to play on them :D 'BT's Fibonacci Sequence'
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Message 1208274 - Posted: 20 Mar 2012, 22:47:44 UTC - in response to Message 1208175.

Nice respondence to Fibonacci numbers thanks.
I am not a math wize I took my math in the old days before computers and we use slide rules so what does a:=1 mean
As a part of the symbol sometimes used to mean "A is defined as 1" or is a=1?
And what does === mean?

On the software I was using, well known constants such as Pi cannot be redefined. If I type "evalf(Pi);", it returns 10 digits (or whatever I set it for) of 3.141592654... .
If I want to define Sarge's constant as 2.176218, i could use SC:=2.176218. So, indeed, the colon before the equals sign indicates I am defining something the software package does not already know to be equal.

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Message 1208313 - Posted: 21 Mar 2012, 1:10:40 UTC - in response to Message 1208274.

Pi is an infinite series. The nth Fibonacci number has a closed form equation. What's interesting is that the equation has irrational numbers in it yet gives a whole number (rational)

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Message 1208334 - Posted: 21 Mar 2012, 1:52:31 UTC
Last modified: 21 Mar 2012, 1:56:14 UTC

I was watching the Science channel on TV and the show was about alien contact and they were discussing seti and the aliens would communicate through math and it would probally be Fibonacci numbers. So I had a program that will run Pi numers to when ever I stop the program so I was wondering about Fibonacci numbers. So I thought I would make this posting and see what the seti crunchers would say.
1+1=2
2+1=3
3+2=5
5+3=8
etc.
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Message 1208349 - Posted: 21 Mar 2012, 2:38:51 UTC - in response to Message 1208313.

Pi is an infinite series. The nth Fibonacci number has a closed form equation. What's interesting is that the equation has irrational numbers in it yet gives a whole number (rational)

That is one definition of Pi.
One could also define it as 4 * integral(sqrt(1 - x^2),0 .. 1,dx).
So many ways to start.

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Message 1208403 - Posted: 21 Mar 2012, 9:21:02 UTC

You also need to look at Lucas Numbers,

2,1,3,4,7,11,18,.......

and Benford's Law which deals with the initial number.

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Message boards : Science (non-SETI) : Fibonacci numbers

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