Category Archives: sequences

Fibonacci multiples, solution 1

Posted on June 9, 2012 by

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 , , , | 5 Comments

A Fibonacci pattern

Posted on May 13, 2011 by

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 , , | 4 Comments

m-bracelets

Posted on November 22, 2009 by

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

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 , , , | 12 Comments

The hyperbinary sequence and the Calkin-Wilf tree

Posted on October 18, 2009 by

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 , , , , , , | 6 Comments

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

Posted on October 12, 2009 by

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 , , , | 4 Comments

More hyperbinary fun

Posted on September 28, 2009 by

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 II

Posted on September 19, 2009 by

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 , | 15 Comments

Predicting pi: pretty graphs and convergents

Posted on January 16, 2009 by

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

Posted on October 23, 2008 by

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