I have a very messly defined function f(n) over naturals s.t. f(2) = 0.01 and it has a recursive relation for n>2:
f(n) = f(n-1) + (f(n-1)+n-1)/10^(2^n)
But everything inside this relation is in base n (so 10=n for example) and the value of f(n-1) is taken in such a way that it looks identical in base n to its n-1 version (for example, f(2) = 0.01 in binary (1/4 in base-10) and within the expression of f(3), f(2) transforms into 0.01 in base-3 (1/9 in base-10)).
So the question is, can this recursive relation be written without using base conversion and in a more algebraic way?
First few values of this function:
0
0.01 in binary (1/4 in dec)
0.010201 in ternary (100/729 in dec)
0.01020103010201 in base-4 (18952481/268435456 in dec)
etc.