# Category Archives: recursion

## More factorization diagrams

My post on factorization diagrams from a month ago turned out to be (unexpectedly) quite popular! I got ten times as many hits as usual the day it was published, and since then quite a few other people have created … Continue reading

Posted in arithmetic, links, pictures, primes, programming, recursion | Tagged , , | 15 Comments

## Factorization diagrams

In an idle moment a while ago I wrote a program to generate "factorization diagrams". Here’s 700: It’s easy to see (I hope), just by looking at the arrangement of dots, that there are in total. Here’s how I did … Continue reading

Posted in arithmetic, pictures, primes, programming, recursion | Tagged , , | 72 Comments

## Post without words #3

(This is my 200th post! =)

Posted in counting, pattern, pictures, posts without words, recursion | 11 Comments

## The hyperbinary sequence and the Calkin-Wilf tree

And now, the amazing conclusion to this series of posts on Neil Calkin and Herbert Wilf’s paper, Recounting the Rationals, and the answers to all the questions about the hyperbinary sequence. Hold on to your hats! The Calkin-Wilf Tree First, … Continue reading

## More hyperbinary fun

When I originally posed Challenge #12, a certain Dave posted a series of comments with some explorations and partial solutions to part II (the hyperbinary sequence). Although I gave the “solution” in my last post, no solution to any problem … Continue reading

Posted in challenges, induction, pattern, proof, recursion, sequences, solutions | Tagged , | 12 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 III

First, a quick recap: continuing an exposition of the paper Recounting the Rationals, we’re investigating the tree of fractions shown below (known as the Calkin-Wilf tree), which is constructed by placing 1/1 at the root node, and giving each node … Continue reading

Posted in number theory, pattern, proof, recursion | 10 Comments