1. **Problem:** Determine a generating function for the sequence $a_r$ given by the number of integer solutions to the equation $$e_1 + e_2 = r$$ with $$e_1 \in \{0,3,4,8\}$$ and $$e_2 \in \{0,4,5,8\}$$. 2. **Formula and Explanation:** The generating function for a sequence counting the number of solutions to $$e_1 + e_2 = r$$ where $$e_1$$ and $$e_2$$ take values from finite sets is the product of the generating functions for each variable. For $$e_1$$, the generating function is: $$G_1(x) = x^0 + x^3 + x^4 + x^8$$ For $$e_2$$, the generating function is: $$G_2(x) = x^0 + x^4 + x^5 + x^8$$ The combined generating function is: $$G(x) = G_1(x) \times G_2(x) = (1 + x^3 + x^4 + x^8)(1 + x^4 + x^5 + x^8)$$ 3. **Intermediate Work:** Multiply the two polynomials: $$\begin{aligned} G(x) &= (1 + x^3 + x^4 + x^8)(1 + x^4 + x^5 + x^8) \\ &= 1(1 + x^4 + x^5 + x^8) + x^3(1 + x^4 + x^5 + x^8) + x^4(1 + x^4 + x^5 + x^8) + x^8(1 + x^4 + x^5 + x^8) \\ &= (1 + x^4 + x^5 + x^8) + (x^3 + x^7 + x^8 + x^{11}) + (x^4 + x^8 + x^9 + x^{12}) + (x^8 + x^{12} + x^{13} + x^{16}) \end{aligned}$$ 4. **Simplify by combining like terms:** $$\begin{aligned} G(x) &= 1 + x^3 + 2x^4 + x^5 + x^7 + 3x^8 + x^9 + x^{11} + 2x^{12} + x^{13} + x^{16} \end{aligned}$$ 5. **Interpretation:** The coefficient of $$x^r$$ in $$G(x)$$ gives the number of integer solutions to $$e_1 + e_2 = r$$ with the given constraints. **Final answer:** $$\boxed{G(x) = (1 + x^3 + x^4 + x^8)(1 + x^4 + x^5 + x^8)}$$ This generating function fully characterizes the sequence $a_r$.