# Tagged Questions

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### How to deduce a closed formula given an equivalent recursive one?

I know how to prove that a closed formula is equivalent to a recursive one with induction, but what about ways of deducing the closed form initially? For example: $$f(n) = 2 f(n-1) + 1$$ I know ...
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### Closed form for recurrence relation

Is there a closed-form solution to the following recurrence: $$T(n) = T(n-1) + T(n-3)$$ If yes, what is it and how can it be proven/derived? If not, then why because a somewhat similar recurrence ...
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### Expressing a Recursion in terms of factorials

Given the recursion $$f(n) = nf(n-1) + (n-1)f(n-2)$$ $$f(0) = 1, f(1) = 1$$ How exactly does one express the target function? I know that $$f(n) = nf(n-1)$$ gives rise to $$f(n) = \Gamma(n+1)$$ ...
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### How to calculate $\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$?
I need to calculate the sum $\displaystyle S=\sum_{n=1}^\infty\frac{(-1)^n}n H_n^2$, where $\displaystyle H_n=\sum\limits_{m=1}^n\frac1m$. Using a CAS I found that $S=\lim\limits_{k\to\infty}s_k$ ...