Articles of recursion

OEIS A000255 recursion.

I encountered the sequence A000255. $a(n)$ counts permutations of $[1,…,n+1]$ having no substring $[k,k+1]$ I am finding difficulty in proving it. Can you please give any clues or hints on how to attack the problem?

Identity involving a recursive product

Here is yet another problem related to plane partitions. Given the recursive formula $$ \begin{align*} F(0)&=1,\\ F(r)&=\prod_{i=1}^r\frac{i+2r-1}{2i+r-2}F(r-1). \end{align*} $$ How can we prove $$F(n)=\prod_{1\leq i\leq j\leq k\leq n}\frac{i+j+k-1}{i+j+k-2}\ ?$$ EDIT: The solution to this problem can be found in the answer section to this question.

Help with generating functions.

Background. Let $P_0(y)=2y-3$ and define recursively $$P_{n+1}(y)=4y\cdot P_n'(y)+(5-4y)\cdot P_n(y).$$ I would like to know as many properties of $P_n$ as I can. For example, it can be shown that each $P_n$ has only real simple positive zeros and that $P_n$ and $P_{n+1}$ strictly interlace for every $n$. It can also be shown that the recursive […]

Proving convergence of a sequence

Let the following recursively defined sequence: $a_{n+1}=\frac{1}{2} a_n +2,$ $a_1=\dfrac{1}{2}$. Prove that $a_n$ converges to 4 by subtracting 4 from both sides. When I do that, I get: $2(\frac{1}{2} a_{n+1} -2)=(\frac{1}{2} a_n -2)$, so $y=2y$, which is true only for $0$. But I’m not sure how to formally use this in a definition of convergence?

Do we really need the recursion theorem if we deal only with specific recursively defined functions?

How is it possible to define in a totally rigorous (i.e. from the axioms) was the functions $$h:\mathbb{N}\rightarrow \mathbb{N}, \ n\mapsto 1\cdot\ldots \cdot n$$ or $$ g:\mathbb{N}\rightarrow \mathbb{R}, \ n\mapsto x^n \ $$ without using the recursion theorem , that immediately tells us that these functions exist and are unique ? Differently said: Do we […]

Derive recursion formula for an integral

Im having trouble understanding questions involving deriving a recursion formula. I need to derive the recursion formula for $I_n$ where $n>=2$ $$I_n = \int(x^2-1)^n dx$$ The other questions ive done so far I used the integration by parts and derived a formula where n is decreasing in the integral. I am using the same method […]

Where do the first two numbers of Fibonacci Sequence come from?

This question already has an answer here: Why does the Fibonacci Series start with 0, 1? 5 answers

$ T(n)= T(\log n)+ \mathcal O(1) $ Recurrence Relation

what is the solution of following recurrence relation? $$\begin{align} T(1) &= 1\\ T(n) &= T(\log n) + \mathcal O(1) \end{align}$$ a) $O(log n)$ b) $ O (log^* n) $ c) $ O (log^2 n) $ d) $ O (n / log n) $

Solve this Recursive Integral

How to solve this recursive integral? I do not even begin to understand how to solve this recursive integral. This doesn’t even seem possible? $$H{(x, y)} = \int_{t=0}^{t=2\pi} H(\frac{xt}{2},\frac{xt}{2})*2xt dt$$ This is intriguing me. What approaches exist for solving this? EDIT:$$H{(x)} = \int_{t=0}^{t=2\pi} H\big(\frac{xt}{2}\big)*2xt \, dt$$

How to derive a closed form of a simple recursion?

Consider: $$T(n) = 2 T(n-1) + 1$$ with $T(1)$ a positive integer constant $a$. I just stuck in finding a closed form for this simple recursion function. I would appreciate it, if someone gives me a hint.