4. A monkey is climbing an 8-rung ladder, each time it can climb one rung or jump two rungs, and at most jump three rungs. From the ground to the top rung, there are $\qquad$ different ways to climb and jump.
4.81.

Starting with simple cases:
(1) If there is 1 step, then there is only one way to climb. That is, $a_{1}=1$.
(2) If there are 2 steps, then there are 2 ways to climb:
(1) Climb one step at a time;
(2) Leap two steps at once, i.e., $a_{2}=2$.
(3) If there are 3 steps, then there are 4 ways to climb:
(1) Climb one step at a time;
(2) Climb one step the first time and leap two steps the second time;
(3) Leap two steps the first time and climb one step the second time;
(4) Leap three steps at once, i.e., $a_{3}=4$.
(4) If there are 4 steps, then we can categorize the ways based on the number of steps taken the first time.
(1) Climb one step the first time, leaving three steps, which according to the previous results, has $a_{3}=4$ ways;
(2) Leap two steps the first time, leaving two steps, which according to the previous results, has $a_{2}=2$ ways;
(3) Leap three steps the first time, leaving one step, which has $a_{1}=1$ way.
Thus, $a_{4}=a_{3}+a_{2}+a_{1}=4+2+1=7$ (ways).
Similarly, $a_{5}=a_{4}+a_{3}+a_{2}=13$ (ways);
$a_{6}=a_{5}+a_{4}+a_{3}=24$ (ways);
$a_{7}=a_{6}+a_{5}+a_{4}=44$ (ways);
$a_{8}=a_{7}+a_{6}+a_{5}=81$ (ways).
Therefore, there are 81 ways to climb.