A method to create a not-decreasing and not-increasing This morning I have explained the monotonous sequences $\{a_n\}$ for my students, but I have had the difficulty of write some simple examples of not-decreasing $a_{n+1}/a_n\geq 1$ and not-increasing sequences $a_{n+1}/a_n\leq 1$. 
How to quickly create examples of not decreasing or not increasing monotonous sequences? Is there a specific criterion or method for to create examples?
 A: Some easy examples of nondecreasing monotonic sequences


*

*The constant sequence:
$$1,1,1,1,1,1,1...$$
Explicit forumula $a_n = 1$.

*The integers repeated once each: 
$$1,1,2,2,3,3,4,4,...$$
Explicit formula: $a_n = \lfloor (n+1)/2\rfloor$

*The partial decimal expansions of $1/27$: 
$$0, 0.03, 0.037, 0.037, 0.03703, ...$$
Explicit formula: $a_n = 10^{-n}\lfloor10^n/27\rfloor$

*The Fibonacci sequence, where the first two terms are $1$ and the remaining terms are the sum of the two previous: 
$$1, 1, 2, 3, 5, 8, 13, 21, 34, 55 ...$$
Explicit formula: $a_n = \lfloor \phi^n/\sqrt{5}\rceil$, where $\phi = (1+\sqrt{5})/2$ is the golden ratio and $\lfloor x\rceil$ is the nearest integer function.

*The number of primes less than $n$:
$$0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, ...$$
Explicit formula: I'd just stick with the primary definition for this one, as it's simple enough to understand why it's not strictly increasing.*


For nonincreasing monotonic sequences, the reciprocals of the above sequences as well as their negations work. I'm having trouble thinking of natural nonincreasing monotonic sequence, though.
*If you must know, it's $a_n =\lceil R(n) - \sum_\rho R(n^\rho)\rceil$, where $R$ is the Riemann $R$ function and the sum runs over all complex roots of the Riemann zeta function.
