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From the above, it is then straightforward to compute the first few
Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …

Example 2.1
Write a program for finding the first 20 Fibonacci numbers.

Solution: The following program fulfills this task:

N=18;
F(1)=1;
F(2)=1;
for k=1:N
F(k+2)=F(k)+F(k+1);
end
F

It should be noted that the value of the different elements of the sequence
depends on the values of the initial conditions, as illustrated in Pb. 2.1, which
follows.

In-Class Exercises

Pb. 2.1 Find the first 20 elements of the sequence that obeys the same recur-
sion relation as that of the Fibonacci numbers, but with the following initial
conditions:

F(1) = 0.5 and F(2) = 1

Pb. 2.2 Find the first 20 elements of the sequence generated by the follow-
ing difference equation:

F(k + 3) = F(k) + F(k + 1) + F(k + 2)

with the following boundary conditions:

F(1) = 1, F(2) = 2, and F(3) = 3

Why do we need to specify three initial conditions?

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