Solving this recursive function $f(x)=f(x-k)+f(x/k)$.

Question

How to solve or simplify the following recursive function? $f(x)$ is defined only for whole numbers as follows:

$$f(x)=\begin{cases} 1 & \mbox{if } x<k; \\ f(x-k)+f(x/k) & \mbox{if } x\ge{k} \mbox{ and } k \mbox{ divides } x;\\ f(x-1) & \mbox{otherwise}. \end{cases}$$

Here $k$ is a positive integer $\le10$.

did you try generating the first terms ? Can you give us some sample ? — Thomas, Nov 03 '13 at 09:36

score 1 · Answer 1 · edited Apr 13 '17 at 12:20

1

This is the number of different ways to write $x$ as the sum of powers of $k$
I found the solution in a 1940 paper, that I copied into the following question: how many ways to make change, asymptotics

edited Apr 13 '17 at 12:20

Community

1

answered Nov 03 '13 at 09:52

Empy2

50,853

Why is it the number of different ways to write x as sum of powers of k ? – kingcobra Nov 03 '13 at 09:55
Can you give a proof for that ? – kingcobra Nov 03 '13 at 09:55
@kingcobra Set up the recurrence. Make it conditional on the existence of a $k^0$ term. Note that $k\leq 10$ is not a required condition. – Calvin Lin Nov 03 '13 at 15:59

Solving this recursive function $f(x)=f(x-k)+f(x/k)$.

1 Answers1