# Tagged Questions

Questions regarding functions defined recursively, such as the Fibonacci sequence.

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### Recurrence relations with multiple roots of auxiliary equation

Suppose we have a homogeneous linear recurrence relation of the form $$u_{n+r}+a_1u_{n+r-1}+\dots +a_ru_n=0$$ The auxiliary equation is then $$f(x)=x^r+a_1x^{r-1}+\dots +a_r=0$$ It is well known that ...
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### Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$

In trying to solve $a^3 - 2b^3 = 1$ over the integers I came across the need to answer the question: when does $(1+ \sqrt[3]{2} + \sqrt[3]{2}^2)^n$ have no $\sqrt[3]{2}^2$ term in it's expansion (in ...
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### How to Find the Recurrence Formula for $\int \frac{dx} {(1+\sin x)^n}$?

$$\int \frac{dx} {(1+\sin x)^n}$$ I know that I should use the following step: $$\int\frac{\sin^2(x)\, dx}{(1+\sin x)^n} +\int\frac{\cos^2(x)\, dx} {(1+\sin x)^n}$$ but here I get stuck. I do ...
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### Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
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### Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
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### number of potential couples

A potential couple is a pair of a man and a woman that like each other (assume that 'like' is a symmetric relation). Given a group of $M$ men and $W$ women, I want to know how many different ...
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