[Solved]: Recursive function calculating number of ways to sum $a + 2 b + 3 c = x$

Recursion is a pretty bad choice here, but here is the recursion you could use: $$ f(n) = begin{cases} 0, & n < 0, 1, & n = 0, f(n-1) + f(n-2) + f(n-3), & n > 0.end{cases} $$ For example, $$ begin{align*} f(-2) &= 0, f(-1) &= 0, f(0) &= 1, f(1) &= f(0) + f(-1) + f(-2) = 1, f(2) &= f(1) + f(0) + f(-1) = 2, f(3) &= f(2) + f(1) + f(0) = 4. end{align*} $$ The dynamic programming approach implied by this example is a much better idea; it can be implemented in constant space and linear time, whereas the recursion will take linear space and exponential time. You could also use matrix powering to compute $f$: $$ f(n) = begin{bmatrix} 1 & 0 & 0 end{bmatrix} begin{bmatrix} 0 & 1 & 0 0 & 0 & 1 1 & 1 & 1 end{bmatrix}^n begin{bmatrix} 1 1 2 end{bmatrix}. $$ The generating function is $$ sum_{n=0}^infty f(n) x^n = frac{1}{1-x-x^2-x^3}. $$ Finally, you can also find an explicit solution: $$ begin{align*} f(n) &= mathrm{round}(Cx^n), C &= frac{1}{3} + frac{sqrt[3]{847+33sqrt{33}}}{66} + frac{4}{3sqrt[3]{847+33sqrt{33}}}, x &= frac{1 + sqrt[3]{19-3sqrt{33}} + sqrt[3]{19+3sqrt{33}}}{3}. end{align*} $$ This explicit solution isn’t very helpful since you need a lot of precision, but it does give you the correct asymptotics; note $C approx 0.6184199223$ and $x approx 1.839286756$.