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