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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

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  • $\begingroup$ HINT: Look up integer sequences in the OEIS. $\endgroup$ – Somos Jun 12 at 13:25
  • $\begingroup$ Thanks Somos that was very helpful. $\endgroup$ – Alex Jun 12 at 14:55
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Thanks to Somos I found the solutions I was looking for. So if anyone else is interested:

$a_n=\frac{1}{16}((2n+3)\cdot\cos(\pi(n-1))+2n^2+6n+3)$

$b_n=\frac{2}{3}(1-\cos(2\pi\frac{n+1}{3}))$

I was specifically looking for solutions that are also differentiable functions over $\mathbb{R}$.

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