# Tagged Questions

Recursion is the process of repeating items in a self-similar way. A recursive definition (or inductive definition) in mathematical logic and computer science is used to define an object in terms of itself. A recursive definition of a function defines values of the functions for some inputs in ...

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### Is there some kind of deep relationship between substitution and recursion?

Define $\mathbb{N}$ as the initial object in the following category: Objects. Sets $X$ equipped with a function $S : X \rightarrow X$ and an element $0:X$. Morphisms. Functions that preserves ...
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### Solve the recursion $p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$

Solve the recursion $p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$ $p_n = p \cdot(1 - p_{n-1}) + (1-p)p_{n-1}$ $= p + (1-2p)p_{n-1}$ I can see that this step simply rearranges the expression, but what'...
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### linear, homogeneous recursion, biological interpretation

Given the recursion $u_{k+1}=\frac{1}{2}(3 u_k - u_{k-1})$ find the expression of $u_k$ in dependence on the values $u_0$ and $u_1$. What is the limit as $k\rightarrow \infty$ of $\{u_k\}_{k}$? Give ...
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### Cutting cheese into chunks [duplicate]

Into how many chunks can one cut a round piece of cheese with n straight cuts? Consider the $3D$ version My try: f(x) = number of pieces and $'x'$ as number of cuts. $f(1)=2$ $f(2)= 2 + f(1)$ ...
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### What is $\lim\limits_{n \to \infty} x_n$ where $x_1=2$ and $x_{n+1}=123+\sqrt{4+5 \sin(x_n) + 6\sqrt{x_n}}$?

$$x_{n+1}=123+\sqrt{4+5 \sin(x_n) + 6\sqrt{x_n}}, \quad x_1=2$$ At first glance, this sequence seems like it will diverge, since it seems like every term is growing by at least $123$. However, I ...
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### If $a_1=1/2$ and $a_{n+1} = a_n^2$, the sequence is convergent

If $a_1=1/2$ and $a_{n+1} = a_n^2$, prove that this recursive sequence is convergent. I know I need to show that it is bounded and monotone decreasing, but I'm not sure how to go about doing that.
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### Closed Form Solution to Exponential Recursion

Is there a closed form solution to the function $f_n=2^{f_{n-1}}$ where $f_0=2$ ? For instance, the first few values of the function are 2, 4, 16, 65536.
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### Finding Height, Number of Leaves, and Value at of Each Node on Recursive Trees

I have an exam tomorrow and am struggling to understand how to get the height of a tree, the number of leaves, and the value of each node. The image is a practice exam. Any tips and help on the first ...
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### How would I find the characteristic equation of this Recurrence Relation?

Find and solve a recurrence relation for the number of $n$-digit ternary sequences with no consecutive digits being equal. Since for ternary, meaning only $3$ possible entries for each space, e.g. $0$...
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### What's the real life purpose of Knuth arrows?

I recently read about Knuth's Arrows. Didn't even know those operations existed. My questions is: Do they have real-life applications? Most of the times a mathematical development follows a real-life ...
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### Converting Non-linear Recursive Series into Explicit Form

I know it's possible to convert any (I think it's any, at least) first-order recursion into an explicit form. For example (assuming I did this right): ...
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### Probability Method Of Recursive number patterns in everyday things

I'd like to know how to go about calculating the probability of a recursive number pattern happening in everyday things for an essay which I'm writing. The example of a recursive number pattern would ...
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### Fractal fundamentals

I am a programmer by trade, and am very interested in fractals. To be very basic about the concept, one might say a 'circle of circles' is a fractal. Where each circle is made up of circles, and ...
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### Plotting y=x^(1/y)

So I was toying around with the idea of recursive functions, where there are two variables, and one of them is on both sides of the equation. I stumbled upon/came up with this function: y = x^(1/y), ...
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### A semi-recursive infinite set is the range of some injective recursive total function

The wikipedia article for semi-recursive sets (formally titled "recursively enumerable sets") claims: A set S of natural numbers is called recursively enumerable if there is a partial recursive ...
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### Showing the number of y < x such that xRy is primitive recursive.

Suppose that $Rxy$ is a (primitive) recursive relation. Let the function $\phi$ be defined as follows: $\phi(x)$ = the number of $y < x$ such that $Rxy$. Show that $\phi$ is (...