Introduction

This short note gives a derivation of a set of numbers similar to the Fibonacci numbers. There numbers will be called the Bob numbers or the Bob integers.²These numbers are defined more carefully below. The technique of matrix analysis using matrix diagonalization is used to find a closed form expression for these numbers.

The author has previously written two papers³ around the time frame of 1990 which contain in essence the derivations given here.

It is also shown that powers of the so-called golden ratio $\psi = {1 + \sqrt{5} \over 2}$

$$\psi^n = \left( {1 + \sqrt{5} \over 2} \right) ^{n}$$

approach an exact integer as $n$ gets large.