# Category: Uncategorized

• ## An M.C. Escher-inspired poster

I wanted an excuse to use Harvey Mudd’s large format printer, so I made a movie-sized (27″×40″) poster for my office based on the second term of OEIS sequence A368138(n): $$A368138(2) = 154$$. The idea here is that you have a a collection of tiles like , which you can rotate and mirror; you then…

• ## Triangle Center Patterns

I made a video that illustrates a particularly interesting “discrete state random dynamical system,” which was inspired by a Tweet (and a mistake) that I saw. First, be hypnotized by this video, which I recommend you watch in 4K, and then scroll down to read about the inspiration and the cool math going on under…

• ## How to Make Animated Math GIFs: LaTeX + TikZ

The first animated GIF that I ever made was made with the LaTeX package TikZ and the command line utility ImageMagick. In this post, I’ll give a quick example of how to make a simple GIF that works by layering images with transparent backgrounds on top of each other repeatedly. TikZ code In our first…

• ## XOR Triangles

In this post, I’ll explore the math behind one of my Twitter bots, @xorTriangles. This bot was inspired by the MathOverflow question “Number triangle,” asked by user DSM posted in May 2020. (I gave an overview of my Twitter bots @oeisTriangles in my post “Parity Bitmaps from the OEIS“. And if you want to build…

• ## Robot Walks

I’ve gotten a lot of mathematical inspiration from Project Euler questions, but perhaps the question that has gotten me thinking the most is Project Euler Problem 208: Robot Walks. In this problem, a robot takes steps either to the right or the left, and at each step, it turns $$\frac 15$$ of the way of…

• ## Pour Le Science and the anti-Sum-Product Problem

In March 2021, I got an out-of-the-blue email from OEIS editor Michel Marcus which totally delighted me. He wrote: This afternoon I went to the library.And I was browsing “Pour La Science” the French version of the Scientific American.And here is what I saw. I like the mysterious tone. He included this photo of an…

• ## Zimin Words and Bifixes

One of the earliest contributions to the On-Line Encyclopedia of Integer Sequences (OEIS) was a family sequences counting the number of words that begin (or don’t begin) with a palindrome: Let $$f_k(n)$$ be the number of strings of length $$n$$ over a $$k$$-letter alphabet that begin with a nontrivial palindrome” for various values of $$k$$.…

• ## Richard Guy’s Partition Sequence

Neil Sloane is the founder of the On-Line Encyclopedia of Integer Sequences (OEIS). Every year or so, he gives a talk at Rutgers in which he discusses some of his favorite recent sequences. In 2017, he spent some time talking about a 1971 letter that he got from Richard Guy, and some questions that went…

• ## Polytopes with Lattice Coordinates

Problems 21, 66, and 116 in my Open Problem Collection concern polytopes with lattice coordinates—that is, polygons, polyhedra, or higher-dimensional analogs with vertices the square or triangular grids. (In higher dimensions, I’m most interested in the $$n$$-dimensional integer lattice and the $$n$$-simplex honeycomb). This was largely inspired by one of my favorite mathematical facts: given…

• ## Parity Bitmaps from the OEIS

My friend Alec Jones and I wrote a Python script that takes a two-dimensional sequence in the On-Line Encyclopedia of Integer Sequences and uses it to create a one-bit-per-pixel (1BPP) “parity bitmaps“. The program is simple: it colors a given pixel is black or white depending on whether the corresponding value is even or odd.…