Counting binary strings of length $n$ that contain no two adjacent blocks of 1s of the same length? Is it possible to count exactly the number of binary strings of length $n$ that contain no two adjacent blocks of 1s of the same length? More precisely, if we represent the string as $0^{x_1}1^{y_1}0^{x_2}1^{y_2}\cdots 0^{x_{k-1}}1^{y_{k-1}}0^{x_k}$ where all $x_i,y_i \geq 1$ (except perhaps $x_1$ and $x_k$ which might be zero if the string starts or ends with a block of 1's), we should count a string as valid if $y_i\neq y_{i+1}$ for every $1\leq i \leq k-2$.
Positive examples : 1101011 (block sizes are 2-1-2), 00011001011 (block sizes are 2-1-2), 1001100011101 (block sizes are 1-2-3-1)
Negative examples : 1100011 (block sizes are 2-2), 0001010011 (block sizes are 1-1-2), 1101011011 (block sizes are 2-1-2-2)
The sequence for the first $16$ integers $n$ is:  2, 4, 7, 13, 24, 45, 83, 154, 285, 528, 979, 1815, 3364, 6235, 11555, 21414. For $n=3$, only the string 101 is invalid, whereas for $n=4$, the invalid strings are 1010, 0101 and 1001.

An aproximation for large $n$
The runs of $0$s and $1$s can be approximated by iid geometric random variables (with $p=1/2$, mean $2$). Hence we have in average $n/2$ runs, of which $n/4$ are runs of $1$s.
Then, the problem is asymtpotically equivalent to : given $m=n/4$ iid Geometric variables $X_1, X_2 \cdots X_m$ find $P_m=$ probability that $X_{i+1} \ne X_i$ for all $i$.
This does not seem a trivial problem, though (and I haven't found any reference).
A crude aproximation would be to assume that the events $X_{i+1} \ne X_i$ are independent. Under this assumption we get
$$P_m \approx P_2^{m-1}= (2/3)^{m-1} \tag 1$$
This approximation is not justified, and it does not seem to improve with $n$ increasing.
The exact value can be obtained by a recursion on the probabilities for each final value, which together with a GF gives me this recursion :
$$P_m = r(1,m) $$
$$r(z,m)= \frac{1}{2z-1} r(1,m-1) - r(2z,m-1) \tag 2$$
with the initial value $r(z,1)=\frac{1}{2z-1}$
Finally, the total number of valid sequences is $C_m = P_m \, 2^n$ ($n=4m$)
I've not yet found an explicit or asympotic for $(2)$.
Some values oc $C_m$
n    m  r(2)            iid(1)          exact
4    1  16              16              13
8    2  170.6           170.6           154
12   3  1950.5          1820.4          1815
16   4  21637.3         19418.1         21414
20   5  243540.2        207126.1        252680     
24   6  2720810.9       2209345.3       2981452
28   7  30515606.3      23566350.0      35179282

