What are some easy to compute irrational/transcendental numbers

Question

I recently wrote a tool that creates fractals out of irrational numbers, in addition to just pretty images. I haven't had the time or cpu to compute their dimensions but they are in the $(1,2)$ interval. Currently I can only really check it with $\pi$ because I found an API happy to serve me the first 7.5 billion digits in groups of 100. I'll attach a link to the code below for anyone interested.

I'd love to know some simple transcendental numbers that I could compute several digits at a time, either with a drip formula or by increasing my approximation accuracy in a sequence approximation.

I've already look at the classics (ex: Euler's Number / Gamma Consant and Liouville's number), but they are either too hard to compute or would be quite bland if drawn in my style (these examples are representative respectively).

Thanks!

Here is an image (5000px, 5000px) of $\pi$ to a couple million digits:

https://cdn.glitch.com/fcea0014-a2c2-415a-bac1-600e666a0f12%2Foverflow_pic.png?1528832361764

Thumbnail (this is quite scaled down to comply with filesize limits:

Source code: https://glitch.com/edit/#!/magical-beard?path=script.js

Live page: https://magical-beard.glitch.me/

Answer 1

Here are some:

• Champernowne's constant $C = 0.12345678910111213141516\ldots$, i.e., the concatenation of the natural numbers in base $10$,
• The Copeland–Erdős constant $C = 0.235711131719232931374143\ldots$, .i.e., the concatenation of primes,
• Liouville's constant: $$ \sum_{n \geq 1} \dfrac 1 {10^{n!} } = \frac 1 {10^1} + \frac 1 {10^2} + \frac 1 {10^6} + \frac 1 {10^{24} } + \cdots = 0.110001000000000000000001000 \ldots $$