Definition of Bob numbers

The so-called Bob integers are defined as the sequence of numbers 1, 3, 4, 7, 11, and so on. The next number is obtained as the sum of the previous two numbers. If the $n^{th}$ number is defined as $x(n)$ then this relation is described by

$$x(n+2) = x(n+1) + x(n) \; , \eqno(1)$$

with

$$x(0) = 1, \; \; x(1) = 3 \; . \eqno(2)$$

This equation is solved in closed form in the next section.