# Number visualization

The following pictures are visualizations of integer sequences. The algorithm used is extremely simple; We look at the binary expansion of an integer and put a black pixel for each 1 and a white pixel for each 0.

For example: If we want to visualize the numbers 0, 1, 2, 3, 4, ... we look at the binary expansions 0, 1, 10, 11, 100, ... Then we write them vertically:

  00001
00110
01010


Which in turn results in the first picture shown below.

The first 400 integers (A000027).

The first 400 squares (A000290).

The first 400 cubes (A000578).

The first 400 triangular numbers (A000217).

The first 400 numbers in the Prouhet-Thue-Morse set (A000069).

The first 400 Fibonacci-odd numbers (A003622).

The first 400 primes (A000040).

The first 400 lucky numbers (A000959).

The first 400 Fibonacci numbers (A000045).

Note that our first algorithm uses 64 bits numbers. Because of this, the pictures do not exceed 64 pixels in height. A full picture (made with the same algorithm, but now with the gmp library) can be seen here.

With this new program we also made a picture of the following sequences:

## Ulam's spiral pictures

Here are some number sequences plotted on Ulam's number spiral (note that the center is in the upper-left corner).
The prime numbers, in the fourier domain and an enhanced version.
The lucky numbers, in the fourier domain and an enhanced version.

We made our own plotting program to get the center in the middle.
The prime numbers, in the fourier domain and an enhanced version.
The lucky numbers, in the fourier domain and an enhanced version.

And here we have plotted the powers up to 1000000.

And the triangle numbers.