So I'm not looking for a general solution but rather a specific one. I'm calculating the time complexity of an algorithm and I would want to be able to formulate the growth of the constants with a good-looking formula.
The sequence is as follows: {1, 4, 13, 40, 121, 364...}. I see that the sequence grows as $f(n) = 3 * f(n-1) + 1$, but is there any way to formulate this sequence non-recursively?
You have $f(1)=1$, and $f(n)=3f(n-1)+1$ gives you, via recursive substitution, $f(2)=3f(1)+1$, $f(3)=3^2f(1)+3^1+1$, $f(4)=3^3+3^2+3^1+1$, and generally $f(n)=3^{n-1}f(1)+3^{n-2}+\ldots+3^{1}+1$. Since $f(1)=1$, you have $f(n)=\sum_{i=1}^{n}3^{i-1}$. Multiply the expression by $\frac{-1+3^n}{-1+3^n}$ and simplify to $f(n)=\frac{-1+3^n}{2}$.
Motivated by the growth rate one could expect that $f(n)=a3^n+b$ for some numbers $a,b$. Solving $f(1)=1$ and $f(2)=4$ for $a,b$ yields $a=1/2$ and $b=-1/2$, which works for the rest of the given sequence.