# Category Archives: sequences

## Predicting pi: pretty graphs and convergents

Recall the challenge I posed in a previous post: given the sequence of integers , what can you learn about (assuming you didn’t know anything about it before)? The answer, as explained in another post, is that you can learn … Continue reading

Posted in convergence, famous numbers, pattern, sequences | Tagged , , , | 5 Comments

## Predicting Pi: solution

Now for the solution to the question in my previous post, which asked what you can learn about , given the sequence of integers . Nick Johnson commented: Well, the obvious thing one can learn given just |(10^n)r| is the … Continue reading

Posted in convergence, pattern, sequences, solutions | Tagged , , , , | 5 Comments

## Predicting Pi

Inspired by a recent post over at Foxmaths!, here’s an interesting challenge question for you to think about: Suppose I give you the sequence of integers , and so on, where denotes the greatest integer less than or equal to … Continue reading

Posted in challenges, pattern, sequences | Tagged , , , | 3 Comments

## Challenge #12 solution, part III

And now for the solution to problem #3 from Challenge #12, which asked: how many ways are there to write a positive integer n as a sum of powers of two, with no restrictions on how many powers of two … Continue reading

Posted in counting, pattern, recursion, sequences, solutions | 1 Comment

## Recounting the Rationals, part II (fractions grow on trees!)

Today I’d like to continue my exposition of the paper “Recounting the Rationals”, which I introduced in a previous post. Recall that our goal is to come up with a “nice” list of the positive rational numbers — where by … Continue reading

Posted in infinity, number theory, pattern, recursion, sequences | 22 Comments

## Carnival of Mathematics #23: Haiku Edition

Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be … Continue reading

## Recounting the Rationals, part I

This is the first in a series of posts I’m planning to write on the paper “Recounting the Rationals“, by Neil Calkin and Herbert Wilf, mathematicians at Clemson University and the University of Pennsylvania, respectively. I’m really excited about it, … Continue reading

Posted in counting, infinity, number theory, pattern, sequences | 19 Comments

## Perfect numbers, part I

Today I’d like to talk about perfect numbers, which touch on some clever algebra and some neat topics in number theory. At least, I’ll start talking about them, since it will probably take me three or four posts to say … Continue reading

Posted in famous numbers, number theory, sequences | 3 Comments

## A few related problems…

Here is a collection of interesting math problems. Despite appearances, they all have something in common. Can you figure out what it is? A single pair of baby rabbits is placed on an island. They take one month to grow … Continue reading

Posted in famous numbers, fibonacci, pattern, sequences | 3 Comments

## The Fibonacci sequence: an introduction

The Fibonacci sequence is probably one of the most famous — and most widely written-about — number sequences in all of mathematics. So why write about it, if it’s been written about so much already? Well, because I can, and … Continue reading

Posted in famous numbers, fibonacci, iteration, pattern, sequences | 11 Comments