Note. I want a function that can be graphed on desmos, I know wolframalpha, Python, JS, etc.., may work, but I really would like to find a function that will work with a standard graphing calculator, I also don't want to use step functions or approximations, etc... I really would like a 'closed function I can play around with, use parametrically, and so on...., so keep that in mind, thank you!
I found / came up with a sequence while working on something. The pattern is easy to spot! Please See My Fig., sorry the string of numbers is so big.
The pattern is, despite the length of time it takes to write out, pretty simple but I can't seem to find its function. Can anyone help? Thank you all so much!
We have $$a_1=1-\frac12$$ and in general, $$a_n = a_{n-1}-a_{n-1}\left(\frac12\right)^n=a_{n-1}\left(1-2^{-n}\right),$$ so that $$a_n=\left(1-2^{-1}\right)\cdots\left(1-2^{-n}\right)=\prod_{k=1}^n\left(1-2^{-k}\right)$$
For example, $$a_5=\frac{1\cdot3\cdot7\cdot15\cdot31}{2\cdot4\cdot8\cdot16\cdot32}$$
EDIT
Here are some more examples.
$$\begin{align} a_1&=\frac12=.5\\ a_2&=\frac{1\cdot3}{2\cdot4}=\frac38=.375\\ a_3&=\frac{1\cdot3\cdot7}{2\cdot4\cdot8}=\frac{21}{32}=.328125 \end{align}$$
If this can be simplified any further, I don't see how.
The points are plotted for $n=1,2,\dots,19$.