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Let us suppose the positive integers $a$, $b$ and $n$ with $a<b$. Is it possible to simplify the following sum:

$$2 \left\lfloor \frac{an}{b} \right\rfloor b + \frac{2^2}{3} \sum_{j=1}^{n-2} 3^j \left\lfloor \frac{a(n-j)}{b} \right\rfloor b $$ ?

I've found a related post: Infinite sum of floor functions but this seems quite different.

Thank you in advance.

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1 Answer 1

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Hint:

Let us leave on the side the terms and coefficient that cause no problem, to focus on

$$\sum b\left\lfloor\frac abk\right\rfloor 3^k.$$

We have $$b\left\lfloor\frac abk\right\rfloor=ak-b\left\{\frac abk\right\}=ak-ak\bmod b.$$

For the left term, which is dominant, there is a closed formula:

$$S=\sum_{k=1}^nkr^k,$$ then

$$S=\sum_{k=0}^{n-1}(k+1)r^{k+1}=r+\sum_{k=1}^n(k+1)r^{k+1}-(n+1)r^{n+1}\\ =r+rS+\sum_{k=1}^nr^{k+1}-(n+1)r^{n+1}\\ =r+rS+\frac{r^{n+2}-r^2}{r-1}-(n+1)r^{n+1}$$ from which you can deduce $S$.

For the right term, $ak\bmod b$ takes all values in range $[0,b)$, scrambled in some order, and repeating peridodically. If $n$ exceeds $b$, full periods will appear, allowing to form groups

$$G_0=\sum_{k=0}^{b-1}(ak\bmod b)r^k,$$ then

$$G_b=\sum_{k=b}^{2b-1}(ak\bmod b)r^k=\sum_{k=0}^{b-1}(ak\bmod b)r^{k+b}=r^bG_0,$$

and more generally

$$G_{mb}=r^{mb}G_0$$ that forms a geometric progression.

For the remaining terms not filling a whole period, there is nothing much better you can do than keeping the original sum.

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  • $\begingroup$ Thanks very much Yves. This is very interesting. I have some questions. 1) Why you consider $\sum b\lfloor\frac abk\rfloor 3^k$? Since the exponent of $3$ shoud be different from $k$. 2) To what exactly refers the closed formula $S$? Thanks again. $\endgroup$
    – Dingo13
    Jul 25, 2016 at 16:20
  • $\begingroup$ @Dingo13: 1) Sorry I don't get your question. 2) $S$ is for the sum of $k3^k$. $\endgroup$
    – user65203
    Jul 25, 2016 at 16:42
  • $\begingroup$ Thanks @YavesDaoust My first question was the following: My sum is $\sum_{j=1}^{n-2} 3^j \left\lfloor \frac{a(n-j)}{b} \right\rfloor b$. Your sum is $\sum_{k=1}^n b\lfloor\frac abk\rfloor 3^k$. They are different, but the results are the same? I have $j$ and $n-j$, and you $k$ at both sides. $\endgroup$
    – Dingo13
    Jul 25, 2016 at 18:07
  • $\begingroup$ @Dingo13: just reverse the terms, this is a minor change. $\endgroup$
    – user65203
    Jul 25, 2016 at 18:51

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