Can you help me to understand better the code I use in my drawings?

Question

I'm a visual artist and I make process drawings using a simple, 3 digit code. the code can be very short or very long, but the number of units is always set - if you take the series of lengths of the code, you get this:
3, 7, 23, 87, etc.
i can see the relationship between these numbers (the value of the new number equals that of the previous one, plus 4 to the power of the ordinal value of the previous one), but I want to know whether this is sequence that comes up anywhere else... and how can I express this in a proper way?

Answer 1

Well, you're saying

N(1) = 3

For k > 1, N(k) = N(k-1)+4^(k-1).

That's equivalent to N(k) = 2 + (1 + 4 + 4^2 + ...) = 2 + (4^k -1)/3

Really. Plug k = 1, 2, 3, 4 into that formula and you get 3, 7, 23, 87.