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Category Archives: sequences
The Recamán sequence
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 selfreference is (for … Continue reading
Posted in arithmetic, recursion, sequences
Tagged difference, integers, Recamán, repeat, sequence
5 Comments
Post without words #5, explained
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, ProuhetThueMorse, sequence, spacefilling, ThueMorse
4 Comments
Fibonacci multiples, solution 1
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
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
mbracelets
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
Recently I’ve been volunteering with the middle school math club at Penn Alexander, a PreK8 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 CalkinWilf 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 CalkinWilf Tree First, … Continue reading
Posted in arithmetic, computation, induction, iteration, number theory, pattern, proof, recursion, sequences, solutions
Tagged algorithm, binary, CalkinWilf, Euclidean, Haskell, hyperbinary, tree
6 Comments
Hyperbinary conjecture seeking proof for a good time, long walks on the beach
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
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
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