\[

a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}

\]

Explanation:

This is 3 times the standard Fibonacci sequence.

Each term is the sum of the two previous terms, but starting with 3,3 , instead of 1,1 .

The standard Fibonnaci sequence starts:

\[

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987, \ldots

\]

The terms of the Fibonacci sequence can be defined iteratively as:

\[

\begin{aligned}

&F_{1}=1 \\

&F_{2}=1 \\

&F_{n+2}=F_{n}+F_{n+1}

\end{aligned}

\]

The general term can also be expressed by a formula:

\[

F_{n}=\frac{\phi^{n}-(-\phi)^{-n}}{\sqrt{5}}

\]

where \(\phi=\frac{1}{2}+\frac{\sqrt{5}}{2} \approx 1.618033988\)

So the formula for a term of our example sequence can be written:

\[

a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}

\]