Generalized Bob Number Representations

A critical assumption in the above analysis and the assumption that results in the $x(n)$ being integers is the assumption on the initial conditions. In the previous section, the initial condition values were $1$ and $3$. In this section, the analysis of the previous section is generalized to the case where the starting values are any arbitrary integers $n_1$ and $n_2$.

$$v(0) = \left[ \begin{array}{c} n_2 \\ n_1 \end{array}\right] \; . \eqno(18)$$

Evaluating Eq. 11 and repeating the above analysis produces the generalized representation

$$x(n) = \left( {1 + \sqrt{5} \over 2} \right) ^{n} \left[ n... ...r 10} \right) - {n_2 \over \sqrt{5}} \right] \; . \eqno(19)$$

Performing steps similar to those associated with Eq. 17 and noting that $x(n)$ is integer for all $n$ shows that the quantity

$$\left( {1 + \sqrt{5} \over 2} \right) ^{n} \left[ n_1 \lef... ... \over 10} \right) + {n_2 \over \sqrt{5}} \right] \eqno(20)$$

exponentially approaches an integer as $n$ gets large. This is true for all integer $n_1$ and $n_2$.

A special case of Eq. 19 is when $n_1 = 0$ and $n_2 = 1$, resulting in the well known Fibonacci sequence $F_0=0, \; F_1=1, \; F_2=1, \; F_3=2, \; F_4=3, \; F_5=5, \; \ldots$ . Using the notation $F_n$ to denote the $n^{th}$ Fibonacci number, Eq. 19 becomes

$$F_n = {1 \over \sqrt{5}} \left[ \left( {1 + \sqrt{5} \over 2... ...ft( {1 - \sqrt{5} \over 2} \right) ^{n} \right] \; . \eqno(21)$$

This is also an elegant form.

Finally, the $F_n$ symbols can be used to express the generalized $x(n)$ numbers of Eq. 19. First we note that the numbers begin as $x(0) = n_1, \; x(1) = n_2, \; x(2) = n_1 + n_2, \; x(3) = n_1 + 2 n_2, \; x(4) = 2 n_1 + 3 n_2, \; \ldots$ and it is seen that the coefficients of the $n_1$ and $n_2$ are the Fibonacci numbers $F_n$ themselves. This relation is stated as

$$x(n+1) = F_n \; n_1 + F_{n+1} \; n_2 \; . \eqno(22)$$

This is easily proved by induction on $n$ using the Fibonacci relation $F_{n+2} = F_{n+1} + F_n$ as well as Eq. 1. This result alternatively follows by using Eq. 21 to give the coefficients in Eq. 19.