Category Archives: sequences

The Recamán sequence

Posted on June 12, 2016 by Brent

I recently learned about a really interesting sequence of integers, called the Recamán sequence (it’s sequence A005132 in the Online Encyclopedia of Integer Sequences). It is very simple to define, but the resulting complexity shows how powerful self-reference is (for … Continue reading →

Posted in arithmetic, recursion, sequences | Tagged difference, integers, Recamán, repeat, sequence | 5 Comments

Post without words #5, explained

Posted on February 7, 2016 by Brent

If you stared for a while at the images in my previous post, you probably noticed some patterns, and maybe you even figured out some sort of rule or algorithm behind them. Commenter Yammatak expressed it as “You split it … Continue reading →

Posted in pattern, pictures, posts without words, sequences, solutions | Tagged curve, Hilbert, Prouhet-Thue-Morse, sequence, space-filling, Thue-Morse | 4 Comments

Fibonacci multiples, solution 1

Posted on June 9, 2012 by Brent

In a previous post, I challenged you to prove If evenly divides , then evenly divides , where denotes the th Fibonacci number (). Here’s one fairly elementary proof (though it certainly has a few twists!). Pick some arbitrary and … Continue reading →

Posted in fibonacci, modular arithmetic, number theory, pattern, pictures, proof, sequences | Tagged divisibility, fibonacci, proof, remainders | 5 Comments

A Fibonacci pattern

Posted on May 13, 2011 by Brent

Recall the Fibonacci numbers, , the sequence of numbers beginning with where each subsequent number is the sum of the previous two: Try this: pick any Fibonacci number. Square it. Now, look at the two Fibonacci numbers on either side … Continue reading →

Posted in algebra, arithmetic, challenges, fibonacci, pattern, sequences | Tagged fibonacci, number, pattern | 4 Comments

m-bracelets

Posted on November 22, 2009 by Brent

It is easy to generalize number bracelets to moduli other than 10—at each step, add the two previous numbers and take the remainder of the result when divided by m. Here are some pretty pictures I made of the resulting … Continue reading →

Posted in arithmetic, fibonacci, iteration, pattern, pictures, sequences | 6 Comments

Number bracelets

Posted on November 17, 2009 by Brent

Recently I’ve been volunteering with the middle school math club at Penn Alexander, a PreK-8 school in my neighborhood. Today we did (among other things) a fun activity I’d never seen before, called “number bracelets”. The students seemed to enjoy … Continue reading →

Posted in arithmetic, iteration, pattern, sequences, teaching | Tagged activity, bracelets, number, Penn Alexander | 12 Comments

The hyperbinary sequence and the Calkin-Wilf tree

Posted on October 18, 2009 by Brent

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 →

Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions | Tagged algorithm, binary, Calkin-Wilf, Euclidean, Haskell, hyperbinary, tree | 6 Comments

Hyperbinary conjecture seeking proof for a good time, long walks on the beach

Posted on October 12, 2009 by Brent

Here’s the latest progress on the hyperbinary sequence. We’re trying to figure out the inverse relation of the function : given a particular number , where does it occur in the hyperbinary sequence? That is, what are the values of … Continue reading →

Posted in challenges, pattern, people, proof, sequences | Tagged Euler, hyperbinary, inverse relation, totient | 4 Comments

More hyperbinary fun

Posted on September 28, 2009 by Brent

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 hyperbinary, induction | 12 Comments

Challenge #12 solution, part II

Posted on September 19, 2009 by Brent

Yes, that’s right, that Challenge #12, posted one year, five months, and a day ago. You see, I have this nasty habit of starting things and not finishing them… well, better late than never! Question two of the aforementioned challenge … Continue reading →

Posted in challenges, counting, induction, pattern, sequences, solutions | Tagged binary, hyperbinary | 15 Comments