# Write a recursive formula for the fibonacci sequence and the golden

So we can think of Fib n being defined an all integer values of n, both positive and negative and the Fibonacci series extending infinitely far in both the positive and negative directions.

### Fibonacci phi formula

Quite analogous to the reproduction of rabbits, let us consider the family tree of a bee - so we look at ancestors rather than descendants. I haven't yet found an explanation for this - can you find one? In a simplified reproductive model, a male bee hatches from an unfertilized egg and so he has only one parent, whereas a female hatches from a fertilized egg, and has two parents. Binet's formula for non-integer values of n? Count the leaves, and also count the number of turns around the branch, until you return to a position matching the original leaf but further along the branch. You can see from the tree that bee society is female dominated. Leonardo, who has since come to be known as Fibonacci, became the most celebrated mathematician of the Middle Ages. The Chevalier asks Pascal some questions about plays at dice and cards, and about the proper division of the stakes in an unfinished game. This interval varies randomly but within a certain range according to external conditions, like temperature, availability of nutrients and so on. We get a doubling sequence. Here A is not zero or we just get a linear equation. This apparently innocent little question has as an answer a certain sequence of numbers, known now as the Fibonacci sequence, which has turned out to be one of the most interesting ever written down.

It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. Both numbers will be Fibonacci numbers. Pascal's response is to invent an entirely new branch of mathematics, the theory of probability.

Just in terms of pure mathematics - number theory, geometry and so on - the scope of his idea was so great that an entire professional journal has been devoted to it - the Fibonacci Quarterly.

## Fibonacci for dummies

Notice that in each row, the second number counts the row. Here is the family tree of a typical male bee: Notice that this looks like the bunny chart, but moving backwards in time. Argand Diagrams Writing x,y for a complex numbers suggests we might be able to plot complex numbers on a graph, the x distance being the real part of a complex number and the y height being its complex part. Argand What is really interesting about the Fibonacci sequence is that its pattern of growth in some mysterious way matches the forces controlling growth in a large variety of natural dynamical systems. So what happens if we plot a graph of F n described by Binet's formula, plotting the results on an Argand diagram? Kurt has an excellent 3D version of the spiral that you can rotate on the screen using a Java applet AND one also for the Lucas numbers formula! The other: hidden away in a list of brain-teasers , Fibonacci posed the following question: If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth? Add up the numbers on the various diagonals There are endless variations on this theme. It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. The blue plot is for positive values of n from 0 to 6. Using this approach, we can successively calculate fn for as many generations as we like. Here are some more examples: Branches of the Fibonacci Family Tree.

Such plots are called Argand diagrams after J. Now let's look at another reasonably natural situation where the same sequence "mysteriously" pops up.

## Fibonacci calculation

The blue plot is for positive values of n from 0 to 6. Now for visual convenience draw the triangle left-justified. It has been rediscovered in an astonishing variety of forms, in branches of mathematics way beyond simple arithmetic. His book was a discourse on mathematical methods in commerce, but is now remembered mainly for two contributions, one obviously important at the time and one seemingly insignificant. Count the leaves, and also count the number of turns around the branch, until you return to a position matching the original leaf but further along the branch. But even more fascinating is the surprising appearance of Fibonacci numbers, and their relative ratios, in arenas far removed from the logical structure of mathematics: in Nature and in Art, in classical theories of beauty and proportion. Binet's formula for non-integer values of n? Electrical engineers tend to use j rather than i when writing complex numbers. Fibonacci could not have known about this connection between his rabbits and probability theory - the theory didn't exist until years later.

It also crosses the x axis at the values -8, 5, -3, 2, -1, 1 and 0 corresponding to the Fibonacci numbers F -6F -5F -4F -3F -2F -1 and F 0. What is really interesting about the Fibonacci sequence is that its pattern of growth in some mysterious way matches the forces controlling growth in a large variety of natural dynamical systems.

Blaise Pascal is a young Frenchman, scholar who is torn between his enjoyment of geometry and mathematics and his love for religion and theology. Now in the Fibonacci rabbit situation, there is a lag factor; each pair requires some time to mature. This is because the curve crosses the x axis at 1 and next at 2, so the distance from the origin has doubled, but the next crossing is not at 4 which would mean another doubling as required for a logarithmic spiral but at 5.

Note that the red spiral for negative values of n is NOT an equiangular or logarithmic spiral that we found in sea-shells on the Fibonacci in Nature page. The number of such baby pairs matches the total number of pairs in the previous generation.

### Golden ratio

Electrical engineers tend to use j rather than i when writing complex numbers. But what Fibonacci could not have foreseen was the myriad of applications that these numbers and this method would eventually have. Note how this curve crosses the x axis representing the "real part of the complex number" at the Fibonacci numbers, 0, 1, 2, 3, 5 and 8. This pattern turned out to have an interest and importance far beyond what its creator imagined. These are called linear equations where A and B are, in general, any real numbers. Now for visual convenience draw the triangle left-justified. In his travels throughout the Middle East, he was captivated by the mathematical ideas that had come west from India through the Arabic countries. Here A is not zero or we just get a linear equation. His idea was more fertile than his rabbits. In fact, all the real values are already in the graph along the x axis also called the real axis. So, one amoebas becomes two, two become 4, then 8, 16, 32, and so on.
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