A rectangle of size $n\times1$ is given.
(a) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $2\times1$?
(b) In how many ways the rectangle can be covered with dominoes of size $1\times1$, $2\times1$, and $3\times1$?
(c) In how many ways the rectangle can be covered with dominoes of size $1\times1$, $2\times1$, $3\times1$, and $4\times1$?
(d) In how many ways the rectangle can be covered with dominoes of size $1\times1$ and $3\times1$?
For the sake of simplicity, I suggest denoting these numbers with $A_n$, $B_n$, $C_n$, $D_n$, respectively.
Solution for (a):
It is obvious $A_1=1$ and $A_2=2$. Let's observe $n\times1$ rectangle, and its last domino (from one short side). It can be $1\times1$ and $2\times1$ domino. This means $A_n = A_{n-1} + A_{n-2}$. In other words, $A_n$ are Fibonacci numbers.
(b), (c):
It looks $B_n$ and $C_n$ are so called tribonacci and tetranacci numbers.
For (d), OEIS A000930 delights in the name of Narayana's cows sequence.
Starting with $n=0$:
$1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745$