# How do I write an explicit form for a sequence that's defined differently for odd and even values of $n$?

My specific question is about how I would write a sequence in explicit form, when the sequence has an explicit form for its odd and even terms. So, using notation, if:

$$a_n = \begin{cases} a_{n_1} & n\space \text{even} \\ a_{n_2} & n\space\text{odd} \\ \end{cases}$$

What is an explicit form for $$a_n$$ in terms of $$a_{n_1}$$ and $$a_{n_2}$$? I assume there's an easy answer but I just thought of this on a whim and don't really care enough to spend hours trying to figure out a solution. For that matter, what if we have this case? Suppose $$a_n$$ goes by a different rule not depending on its remainder mod $$2,$$ but its remainder mod $$k$$. So, say it's defined as follows:

$$a_n = \begin{cases} a_{n_1} & n\equiv 0\space(\text{mod}\space k) \\ a_{n_2} & n\equiv 1\space(\text{mod}\space k) \\ .\\ .\\ .\\ a_{n_{k-1}} & n\equiv k-1\space(\text{mod}\space k)\\ \end{cases}$$

Can we get one explicit expression for $$a_n$$?

• If you want, you can use roots of unity (or equivalently sine and cosines) to get a single "formula". But that is pointless and \begin{cases}...\end{cases} is already explicit. Jun 1, 2021 at 5:50
• Perhaps "explicit" wasn't the right word, as you're correct, it is explicit. I should have said "not piecewise" or something to that effect. Jun 1, 2021 at 6:00

One can make use of periodic functions such as trigonometric functions. For example, in the case of $$\hbox{mod }2$$, write $$a_n=|\sin(n\pi)|a_{n_2}+|\cos(n\pi)|a_{n_1}$$. One, can also make use of roots of unity. For example, the same formula can be written as $$a_n=\dfrac{1-(-1)^n}{2}a_{n_2}+\dfrac{1+(-1)^n}{2}a_{n_1}$$. Similarly, for $$\hbox{mod k}$$, try to make use of $$k$$th roots of unity. For $$\hbox{mod } 3$$, note that $$\dfrac{1+{\omega}^n+{\omega}^{2n}}{3}$$ is $$0$$ for $$n \neq 3k, k \in \mathbb{Z}$$ and is $$1$$ for $$n=3k, k \in \mathbb{Z}$$.