AMT C25-29

The more important of these properties have already been proved, but none of them should give the reader any difficulty.

One may need to be concerned with other sequences $p_n$ satisfying the Fibonacci law, e.g. if one is concerned with specimens whose principal parastichy numbers are taken from the sequence 1, 3, 4, 7, 11, ... Although the general term of such a sequence can be expressed in the form $A\omega^n+B(-\omega^{-1})^n$ it really seems more convenient to relate it to the Fibonacci numbers by the equation $p_n=p_0 F_{n-1}+p_1 F_n$.

In connection with various applications, it is convenient to note the following

Wronskian property of solutions of the Fibonacci equation

Suppose

$p_{n+1}=p_{n}+p_{n-1}

(8.10)

q{n+1}=q_{n}+q{n-1}

for each $n$, then for each $n$ also

p_{n+1}q_n - p_n q_{n+1} = (-)^n (p_1 q_0-p_0 q_1) (8.11)

Since

$(p_{n+1}q_n-p_n q_{n+1})+(p_n q_{n-1}-p_{n-1} q_n)=q_n (p_{n+1}-p_n-p_{n-1})- p_n(q_{n+1}=q_n-q_{n-1})=0$

it follows that $(p_{n+1}q_n-p_n _{n+1})(-)^n$ is independent of $n$.

In this and other properties of the Fibonacci equation many readers will notice a similarity to the theory of linear differential equations.