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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Number of sequences of $0$s, $1$s, and $2$s with length $n$ such that there is a $0$ somewhere between every pair of $2$s

Let $a(n)$ be the number of sequences with length $n$ which consists the digits $0,1,2$ such that between every two occurrences of $2$ there is an occurrence of $0$ (not necessarily next to the $2$'s)....
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Functional equation: Show $0\le f(n+1)-f(n)\le 1$ and find all $n$ such that $f(n)=1025$.

The function $f:\mathbb{N}\to \mathbb{R}$ satisfies all of \begin{align}f(1)&=1, \\ f(2)&=2,\\ f(n + 2) &= f(n + 2 − f(n + 1)) + f(n + 1 − f(n)) \tag{1} \end{align} Show ...
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Primitive recursion and $\Delta^0_0$

Until recently I assumed that primitive recursive relations are exactly $\Delta^0_0$ (i.e. bounded) ones, but I learned they're different (the former is a proper superclass of the latter). I have ...
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Can every recursive formula be expressed explicitly?

I'm not sure if my wording is entirely correct, but I was just wondering if every recursive formula can be turned into an explicit formula. I am asking this because various sources online gives me ...
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General overview about Recursion (free online texts)

I'm looking for free online texts about recursion. What I'm looking for formal definitions* of "all" (most of) the different types of recursion and from different points of views like Category ...
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Is there a recursive formula for Euler's Totient function

I have been looking for a recursive formula for Euler's totient function or Möbius' mu function to use these relations and try to create a generating function for these arithmetic functions.
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Precisely, what is a primitive recursive definition?

If I understand correctly, the language of primitive recursive arithmetic has a distinguished function symbol $S$, together with a function symbol for each primitive recursive definition (hereafter ...
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Killer Tree! (one of my old questions)

There is a problem that killed me! but I couldn't solve it: We have a tree graph witch its structure is what is on image. Proof that there is no reduplicative numbers in each line.
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Are hyperoperators primitive recursive?

I apologize if this question is too basic. I have read that the Ackerman function is the first example of a computable but NOT primitive recursive function. Hyperoperators seem to be closely related ...
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The largest root of a recursively defined polynomial

Suppose that for all $x \in \mathbb{R}$, $f_1(x)=x^2$ and for all $k \in \mathbb{N}$, $$f_{k+1}(x) = f_k(x) - f_k'(x) x (1-x).$$ Let $\underline{x}_k$ denote the largest root of $f_k(x)=0$. I ...
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Game in a circle

$N$ players play a game. They stand in a way such that they form a regular $N$-gon. Players are numbered from $1$ to $N$. The players throw boomerangs in clockwise order, in turns. At first player $1$ ...
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undecidability of the structure $(\omega,+,2^n)$

Is the structure $(\omega,+,2^n)$ undecidable? There is no easy way to define multiplication using a formula.
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Prove function is Fibonacci sequence

Stuck on how to finish a question I'm working on. Have to find the number of bitstrings of length n with no odd length maximal runs of ones. For example, when n=3 there are three such bitsrings: 011, ...
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Where do the first two numbers of Fibonacci Sequence come from? [duplicate]

I'm trying to code a simple algorithm that prints out the $n^{th}$ Fibonacci number. However, my program requires the initial seed values $F_0 = 0$ and $F_1 = 1$, when I'm hopeful I can figure ...
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A Recurrence Relation Involving a Square Root

Consider the recurrence relation: $a_{n+1} = \sqrt{a_n^2 -k},$ where $k>0$, $n\in\{0,1,n:a_n^2\geq k\}$, and $a_0>0$ is known. Is it possible to obtain an expression for $a_n$ in terms of $n$?...
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Why is $\log(b,n) = \lfloor \log_b(n) \rfloor$ primitive recursive?

I read in an introduction to primitive recursive function and Wikipedia that $$\log(b,n) = \lfloor \log_b(n) \rfloor$$ is primitive recursive. But how can that be? Is there any easy proof (and ...
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From Primitive Recursive to Recursive by Iterating over more than one Argument?

Is the only way a function can be recursive and not primitive recursive by growing faster than primitive recursion allows (as with Ackerman's function)? If so, then consider the following. Primitive ...
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Alternative unconditional form of $\sqrt{n -\sqrt{n -\sqrt{n -\cdots}}}$?

Consider $a_n$, where \begin{align} a_n &=\small{\sqrt{n -\!\!\!\sqrt{n -\!\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\!\sqrt{n -\!\sqrt{n - \cdots}}}}}}}}\end{align} Using a recursive ...
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If a recursive sequence converges, must its inverse be divergent?

Suppose I have a recursive sequence $\displaystyle a_{n+1} = \frac{a_{n}}{2}$. Clearly, the sequence converges towards zero. Now, suppose I define an "inverse" sequence $\displaystyle b_{n+1} = 2b_{n}$...
The set $F_{n}$ of primitive recursive function symbols of arty $n$ can be defined inductively as \begin{array}[lr] & Z, \text{Succ} \in F_{1} & \\ \pi_{j}^{n} \in F_{n} \quad \text{for each} \...