I'm searching for the name of these kind of sequences:
Suppose you start off with a finite sequence containing one term:
S0 = {3}
To get the next sequence you first tack on a copy the first sequence, then take its elements, add a constant to each element, and tack them on as well to the end:
S(n+1) = {S(n),S(n) + 7}
For example:
S0 = {3}
S1 = {3,10}
S2 = {3,10,10,17}
S4 = {3,10,10,17,10,17,17,24}
etc...
These sequences are formed by a simple recursive pattern, but I can't seem to find a reference to them in my undergraduate textbooks (I only find information on linear recursion and such).
Any help in identifying the name and where I can find more information about their properties is greatly appreciated.
UPDATE: After researching common notations for sequences, I think I've come up with a mathematical representation of the sequences (using $\frown$ to represent concatenation):
$$A_0 = (3)$$ $$A_n = (A_{(n-1)_i})_{i=0..(2^{n-1}-1)} \frown (A_{(n-1)_i} + 7)_{i=0..(2^{n-1}-1)}$$
However, a former professor of mine derived a far better expression:
$$A_0 = 3$$ $$A_{2^n+k} = A_k + 7$$ $$n \ge 0$$ $$0 \le k < 2^n$$
I'm still haven't found out the name of these type of sequences though.