Discrete Math: Recurrences and Generating Functions

Find the coefficient of $x^5$ in the binomial expansion of $(2x - 8)^8$.

Given that the coefficient of $x^3$ is 3 times that of $x^2$ in the expansion $(2 + 3x)^n$, find the value of $n$.

Given a sequence generated by the rule $x_n = x_{n-1} + x_{n-2}$, determine the ratio of consecutive terms as it approaches a limit, and prove that this ratio is the golden ratio $\frac{1 + \sqrt{5}}{2}$ or its negative inverse.

Using generating functions, determine how many ways there are to combine 10 candies when the candies are red, blue, and green with the conditions: even number of red candies, more than six blue candies, and less than three green candies.

Find the number of non-negative solutions to $x_1 + x_2 + x_3 = 6$, where $x_1 \leq 4$, $x_2 \leq 3$, and $x_3 \leq 5$ using generating functions.

Find the generating function for a sequence given recursively by: $a_n = 2a_{n-1} + 4a_{n-2}$ with initial terms $a_0 = 1$, $a_1 = 3$, and $a_2 = 10$.

Find the generating function for a sequence given recursively by: $a_n = a_{n-1} + 2a_{n-2} + 3$ with initial terms $a_0 = 2$ and $a_1 = 2$.

Solve the recurrence relation $a_n = 3a_{n-1} - 2a_{n-2}$ with initial conditions $a_0 = 1$ and $a_1 = 3$.

Define the recurrence relation for a geometric sequence as $a_n = a_{n-1} \times 2$ for $n \geq 1$, starting with $a_0 = 3$.

Solve the second-order linear homogeneous recurrence relation $a_n = 5a_{n-1} - 6a_{n-2}$ with initial conditions.

Given the recursive formula $a_{n+1} = 3a_n + 2$, and the first term $a_1 = 1$, find the next four terms.

Given the recursive formula $a_n = 2(a_{n-1})^2 - 5$, with the first term $a_1 = 2$, find the next three terms.

Given the recursive sequence defined by $a_0 = 1$ and $a_n = 1 + a_{n-1}^2$, compute the first few terms of the sequence.

Using the recursive relation for a sequence similar to the Fibonacci sequence, where $a_0 = 1$, $a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2} + a_{n-3}$, find the first few terms of the sequence.

Using the logistic sequence recursive relation $a_{n+1} = 2 \cdot a_n \cdot (1 - a_n)$, with an initial term between 0 and 1, calculate the behavior of the sequence.

Given a linear homogeneous recurrence relation with initial conditions, solve the recurrence relation using the characteristic equation technique. The example involves getting all terms over to the left side to form the characteristic equation, finding the general form of the solution, and determining constants using initial conditions.

Solve the recurrence relation $T(n) = 2T(n/2) + 4n$ with the initial condition $T(1) = 4$ using the iterative substitution method.

Prove that for the recursively defined sequence where the first term is 1, the second term is 3, and the k-th term is defined as $a_k = a_{k-2} + 2a_{k-1}$, all terms are odd.