One way to do this is with the transfer matrix method. (See, e.g., chapter 4 of Richard Stanley's Enumerative Combinatorics, volume 1.) The basic idea is that as you draw the balls, you keep track of the colors of the last two balls drawn. You can represent the possible transitions as edges in a directed graph with weights that keep track of the number of R's, B's, and triples of each type, so that the numbers you want can be obtained by extracting coefficients from powers of a 4 by 4 matrix, and from this matrix you can get a rational generating function for the numbers.

This identity is also a special case of Vandermonde's theorem.

For an arbitrary Turing machine there is little or nothing that can be said.

More generally, for $p\ne 2$ or 5, $F_{n+p} \equiv F_{n+1} \pmod p$ if $(p|5)=1$ and $F_{n+p}\equiv -F_{n-1} \pmod p$ if $(p|5) = -1$. (Here $F_{-1}=1$.)

The generating function $f(x)=\sum_n n^n x^n/n!$ is equal to $1/(1+W(-x))$, where $W$ is the Lambert $W$-function. The denominator has a zero at $x=1/e$ which should enable you to get a good approximation to the coefficients of $f(x)^k$ by standard techniques (see, e.g., Flajolet and Sedgewick's Analytic Combinatorics).

See Emeric Deutsch, Dyck Path Enumeration, Discrete Mathematics 204 (1999) 167-202, section 6.5, sciencedirect.com/science/article/pii/S0012365X98003719

See also Curtis Greene and Daniel J. Kleitman. Strong versions of Sperner’s theorem. J. Combinatorial Theory Ser. A, 20(1):80–88, 1976. For similar decompositions of other posets, search for "symmetric chain decomposition".

OK, Joyal described his proof using species, but he used (implicitly) a non-natural bijection; more precisely, he used the fact that the number of linear orders of a finite set is equal to the number of permutations (i.e., sets of cycles) of the set, but the corresponding species are not isomorphic. The proof may have been inspired by species, but I don't think it's a good example of what species are good for. There are other ways to deal with exponential generating functions; the real power of species (at least in enumeration) is its application in enumeration under group action.

But Joyal's proof does not use species. His proof may have been inspired by species, but his bijection is not natural (i.e., not functorial) — not that there's anything wrong with that.

This hypergeometric expression says no more and no less than that the coefficient of $x^{k+2}$ in the sum is $$\frac{1}{(n-k)!\,(k+2)}.$$

Alpert's theorem was proved earlier by Clemens Heuberger, "Minimal expansions in redundant number systems: Fibonacci bases and greedy algorithms", Periodica Mathematica Hungarica 49 (2), 2004, 65–89.