**Polya urn model**: At time $0$ an urn initially contains $b$ $\tt{B}$lue balls and $r$ $\tt{R}$ed balls. At time $1$, a ball is drawn uniformly at random (removing it) from the urn, and *two* balls of the drawn color are added to the urn. This procedure is then repeated at times $2,3,4,\ldots$, each time increasing by one the total number of balls in the urn while randomly changing its composition, as follows:

$$(x,\ y) \longrightarrow\begin{cases} (x+1,\ y), & \text{with probability $x\over x+y$} \\[2ex] (x,\ y+1), & \text{with probability $y\over x+y$} \end{cases}$$

where $x$ (resp. $y$) denotes the number of blue (resp. red) balls.

Let $T_w$ denote the time of the first occurrence of word $w\in\{\tt{B,R}\}^*$ in the sequence of drawn colors. (E.g., $T_\epsilon=0$ always, and in the sequence $\tt{RBRBB}\ldots$, $T_{\tt{B}}=T_{\tt{RB}}=2,\ T_{\tt{BB}}=5$, etc.)

Question: Is there a knownclosed formfor $E(T_w)$ in any case other than $T_{\tt \epsilon}$, $T_{\tt B}$, or $T_{\tt R}$? If so, what is it, and how to derive it?

This was asked on MathSE two weeks ago and remains unanswered. As shown there, it is straightforward (via hypergeometric series properties) that for all integers $r\ge 0$ and $b\gt 1$, $$E(T_{\tt{B}})\ =\ {\Large\sum}_\limits{k=0}^\infty{(r)_k(b)_1\over (r+b)_{k+1}}(k+1)\ =\ 1+{r\over b-1},\tag{1}$$ where the Pochhammer symbols are defined as follows: $$(a)_n = \begin{cases} 1 &\text{ if }n = 0 \\ a(a+1)(a+2)\cdots (a+n-1) &\text{ if } n > 0. \end{cases}$$ However, $E(T_{\tt{BB}})$, for example, seems much less tractable: $$\begin{align}E(T_{\tt{BB}})&={\Large\sum}_\limits{j=2}^\infty{\Large\sum}_\limits{i=j-2}^\infty{(r)_i(b)_j\over (r+b)_{i+j}}\binom{i}{j-2}(i+j)\tag{2}\\[2ex] &={\Large\sum}_\limits{i=0}^\infty{\Large\sum}_\limits{j=0}^i{(r)_i(b)_{j+2}\over (r+b)_{i+j+2}}\binom{i}{j}(i+j+2)\tag{3}\\[2ex] &\overset{??}{=}\ \left(1+{r\over b-1}\right)\left(2+{r\over b-2}\right)\quad(r\ge 0,\ b\ge 3).\tag{4} \end{align}$$

Equation (4) is an unproved conjecture resulting from inspecting, for a variety of $(b,r)$ values, simulations of the urn process, as well as numerical evaluations of the double sums (2) and (3).