Fibonacci recurrence

Fig. 4.10. A beam of spin-1/2 particles (arrow) passing through the spin-precession apparatus M4 consisting of five magnets arranged according to a Fibonacci recurrence.

R. P. Brent, On the Periods of Generalized Fibonacci Recurrences, Math. Comp. 63,... 

The Fibonacci multilayer structures belong to the class of deterministic non-periodic media. They are quasiperiodic structures because they include several incommensurate periods. Multilayer binary Fibonacci stacks of n-th stage are recurrently generated by the following rule ... 

Another common example of a recurrence relation is one that defines the Fibonacci numbers. This sequence of numbers, called the Fibonacci sequence after the wealthy Italian merchant Leonardo Fibonacci of Pisa, plays important roles in mathematics and nature. These numbers are such that, after the first two, every number in the sequence equals the sum of the two previous numbers ... 

This program computes the first 30 Fibonacci numbers, using an array size of 30. This method of using arrays to store previous results is usually the preferred method for evaluating recurrence relations, because it allows even complex expressions to be processed in a uniform and efficient manner. Of course, in this example, the programmer can even avoid the array by retaining the last two values in two variables. 

One can easily see that the recurrence defined by Eq. (34) is structurally identical to the recurrence defining the Fibonacci numbers. Indeed, it is enough to set p = tm=l for ali m and initialize this sequence as = 1. The sequence of the Fibonacci numbers may be produced by the diagram shown in the left panel of Fig. 1 and the number of summands needed to compute the coefficient aj is equal to the Fibonacci number... 

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