Is $h(n)$ independent of $n$?

Question

Let $n$ be a positive integer.
Let $f(n) = lfloorfrac{8n+13}{25}rfloor$ and $g(n) = lfloorfrac{n-17}{25}rfloor$.
Now consider $$h(n) = f(n) - lfloorfrac{n - 12 - g(n)}{3}rfloor$$
Then I calculated values of $h(n)$ for various values of $n$ and I'm getting $h(n)=4 $ everytime. Infact I ran a C code also and verified it till $n = 10^6$. But I am not able to prove this equality. How can I prove this?

Amit Rajaraman · Accepted Answer

Let $n=25k+r$ for some non-negative integers $k,r$ with $r<25$. We then have
$$leftlfloorfrac{n-12-leftlfloorfrac{n-17}{25}rightrfloor}{3}rightrfloor = leftlfloorfrac{24k+r-12-leftlfloorfrac{r-17}{25}rightrfloor}{3}rightrfloor = 8k-4+leftlfloorfrac{r-leftlfloorfrac{r-17}{25}rightrfloor}{3}rightrfloor.$$
We also have
$$leftlfloorfrac{8n+13}{25}rightrfloor=8k+leftlfloorfrac{8r+13}{25}rightrfloor.$$
So it suffices to show that for any non-negative integer $r<25$,
$$leftlfloorfrac{8r+13}{25}rightrfloor - leftlfloorfrac{r-leftlfloorfrac{r-17}{25}rightrfloor}{3}rightrfloor=0.$$
Try showing that this is true.

Is $h(n)$ independent of $n$?

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