Fibonacci numbers

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.
Tullio
____________

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/.


Lt

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

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

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.

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

(1 + sqrt(5)) / 2

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

>

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

>

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

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?

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)

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)