let's say we have a sequence has the restricted growth property if it is a sequence of positive integers $a_{1}, a_{2}, a_{3}....a_{n}$ such that:
- $a_{1}=1.$
- $a_{n+1} \leq$ Max$_{1\leq i \leq n}$ ($a_{i}+1$)
What are the exactly steps to calculate $a_{2}$? Sorry if this looks like a stupid question. but would like to receive some help since have not in school for a while. Here is my attempt:
$a_{2} \leq a_{1+1}$, so $a_{2} \leq$ Max$_{1\leq i \leq 1 }$ ($a_{i}+1$)
In this case, i can only be 1. Therefore, $a_{2}$ $\leq$ Max$_{1\leq 1 \leq n}$ ($a_{1}+1$) or simply $a_{2}\leq Max(2).$ What exactly is the sequence ? any example is really appreciated.
There is no way to calculate the exact values of $a_2,a_3,\cdots $ since $a_2\in \{1,2\}$ , $a_3\in \{ 1,2,3\}$ and more generally$$a_n\in \{1,2,\cdots ,n\}$$Here are some examples $$a_n=1\quad,\quad \forall n\in \Bbb N\\a_n=n\quad,\quad \forall n\in \Bbb N\\a_n=\lfloor \sqrt n\rfloor\quad,\quad \forall n\in \Bbb N\\a_n=\lfloor \ln n\rfloor+1\quad,\quad \forall n\in \Bbb N$$