수열 전체 문제

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$, assume that $a_1=1$, $a_2=1$, $a_3=1$, and for that each natural number $n$,

$$a_{n+3} = a_{n+2} + a_{n+1} + a_n$$

.

Compute $a_4$, $a_5$, $a_6$, and $a_7$.

For the sequence $a_1, a_2, \ldots, a_n, \ldots$, assume that $a_1 = 1$, and that for each natural number $n$,

$$a_{n + 1} = a_n + n \cdot n!$$

.

Compute $n!$ for the first 10 natural numbers.

For the sequence $a_1, a_2, \ldots, a_n, \ldots$, assume that $a_1 = 1$, and that for each natural number $n$,

$$a_{n + 1} = a_n + n \cdot n!$$

.

Compute $a_n$ for the first 10 natural numbers.

For the sequence $a_1, a_2, \ldots, a_n, \ldots$, assume that $a_1 = 1$, and that for each natural number $n$,

$$a_{n + 1} = a_n + n \cdot n!$$

.

Make a conjecture about a formula for $a_n$ in terms of $n$ that does not involve a summation or a recursion.