# Sequences with reocurring/doubled elements: closed expression

I'd like to ask for your help in finding the closed-form expressions for the $$n^{th}$$ term of sequences with reoccuring or doubling items. Specifically the following sequences:

$$(a_n)=(0,1,1,3,3,6,6,10,10,15,15,...)$$ where $$a_0=0,$$ and $$a_{2n+1}=a_{2n+2}=\frac{(n+1)(n+2)}{2},~ n=0,1,2,...$$

$$(b_n)=(1,1,0,1,1,0,...)$$ where $$a_{3n}=a_{3n+1}=1,$$ and $$a_{3n+2}=0,~n=0,1,2,...$$

I would like to know a closed expression for
$$a_n=f(n),$$
$$b_n=g(n)$$.

In case you are interested, I am currently working on divergent series and their potential values.

If you can help me that would be much appreciated.

Cheers, Alex

• HINT: Look up integer sequences in the OEIS. – Somos Jun 12 at 13:25
• Thanks Somos that was very helpful. – Alex Jun 12 at 14:55

$$a_n=\frac{1}{16}((2n+3)\cdot\cos(\pi(n-1))+2n^2+6n+3)$$
$$b_n=\frac{2}{3}(1-\cos(2\pi\frac{n+1}{3}))$$
I was specifically looking for solutions that are also differentiable functions over $$\mathbb{R}$$.