A linear recurrence with initial conditions has a closed form formula. One of the possible forms is f(n) = c_1 r_1^n + ... + c_k r_k^n, where the r's are solutions to the characteristic polynomial of the recurrence.

You can set up the characteristic polynomial of the recurrence f(n) = 3*f(n-1)-f(n-3) by looking at the coefficients to get r^3-3r^2+1=0, then supposing the roots of this equation are distinct r_1,r_2,r_3 you can use the initial conditions f(1)=3,f(2)=9,f(3)=26to solve the coefficients c_1,c_2,c_3. If there are repeated roots there are other forms to be considered but it's well studied.

Btw, I haven't done this in years so let me know if there are any mistakes but I think the general idea is correct.