Hello, I intend of course to make an english version of this very short note. The graphic is in false colors, of course there are only 2 colors : white and light blue, depending on the screen and the magnification factor. I presumed that most people have access to a kind of mathematica, maple, or pari-gp or mpmath like program in order to verify that the claim is authentic, when f(n) is evaluated at n=1000000 or 2^20 the pattern pushes up to 2^40 which is 1000 billion binary digits. I made it that way so that we could see the pattern in 1 image. To see the image with whatever system you may have, I use Photoshop which can view an image of up to 90 billion pixels : 300000 x 300000. Here is the recipe : Prepare a binary file or if you want a file containing only <0> and <1> with no CR or LF , only plain binary file. Change the suffix of the file to the_file.raw , in raw format. Open Photoshop and open the file , it will then ask you for the width and length, then type the 2 numbers. It is now necessary to adjust the tone and brightness or contrast of the image. There are then a variety of ways to do that. Now about a possible generating function, well I know a bit about this subject, in plain English : there are no known formulas that would exhibit a pattern up to the billion'th term in terms of analytical or ordinary generating function ans still be visible for an ordinary eye, the example with Catalan numbers is way too fast growing , the n'th term is proportional to 4^n, this is way too big. The billionth Catalan number has hundreds of millions digits wide, it cannot be something simple, the complexity grows exponentially. As I said before, there is a wide grey area where the pattern is clearly visible and still cannot be expressed simply with a simple generating function or rational or a certain distance to irrationals, the pattern is there but cannot be simply explained in analytical terms, how to do you explain that the formula for f(2^20) can still be visible after 1000 billion binary digits ?? Sorry for the crude and french language results, but still the pattern remains and it is still very very persistent. Best regards, Simon Plouffe