Reference Request for solution of Ramanujan Identities Where can I find the identities of Ramanujan concerning the Floor Function with its solution? Any site you can recomend to me? 
 A: For (i), write $n=6k+r$ where $r\in \{0,1,...,5\}$. Then left side is $$\begin{eqnarray}\Big[{n\over 3}\Big]+ \Big[{n+2\over 6}\Big]+\Big[{n+4\over 6}\Big] &= &\Big[{6k+r\over 3}\Big]+ \Big[{6k+r+2\over 6}\Big]+\Big[{6k+r+4\over 6}\Big]\\
&= &2k+\Big[{r\over 3}\Big]+ k+\Big[{r+2\over 6}\Big]+k+\Big[{r+4\over 6}\Big]\\
&= &4k+\underbrace{\Big[{r\over 3}\Big]+\Big[{r+2\over 6}\Big]+\Big[{r+4\over 6}\Big]}_{E_r}\\
\end{eqnarray}$$
$$E_r=\left\{%
\begin{array}{ll}
    0, & r=0,1\\
    1, & r=2 \\
2, & r=3 \\
3, & r=4,5 \\
\end{array}%
\right.$$
And the right side is
$$\begin{eqnarray}\Big[{n\over 2}\Big]+ \Big[{n+3\over 6}\Big]&= &\Big[{6k+r\over 2}\Big]+ \Big[{6k+r+3\over 6}\Big]\\
&= &3k+\Big[{r\over 2}\Big]+ k+\Big[{r+3\over 6}\Big]\\
&= &4k+\underbrace{\Big[{r\over 2}\Big]+\Big[{r+3\over 6}\Big]}_{F_r}\\
\end{eqnarray}$$
$$F_r=\left\{%
\begin{array}{ll}
    0, & r=0,1\\
    1, & r=2 \\
2, & r=3 \\
3, & r=4,5 \\
\end{array}%
\right.$$
So both sides are the same for all $r$ and we are done.
A: If $[x]$ denotes floor we have
$$\sum\limits_{k=0}^{m-1}\Big[{n+ks+t\over ms}\Big]=\Big[{n+t\over s}\Big]$$
for $n\geqslant0$, $m>0$, $s>t\geqslant0$, $n,m,s,t$ - integers. So
$$\Big[{n\over 2}\Big]=\Big[{n\over 6}\Big]+\Big[{n+2\over 6}\Big]+\Big[{n+4\over 6}\Big]$$
$$\Big[{n\over 3}\Big]=\Big[{n\over 6}\Big]+\Big[{n+3\over 6}\Big]$$
and obviously
$$\Big[{n\over 3}\Big]+\Big[{n+2\over 6}\Big]+\Big[{n+4\over 6}\Big]=\Big[{n\over 2}\Big]+\Big[{n+3\over 6}\Big]$$
With same conditions for $n,m,s,t$ if $[x]$ denotes ceiling we have
$$\Big[{n\over 3}\Big]+\Big[{n+2\over 6}\Big]+\Big[{n+4\over 6}\Big]=\Big[{n+2\over 2}\Big]+\Big[{n+3\over 6}\Big]$$
and also if $[x]$ denotes nearest integer we have
$$\Big[{n+2\over 3}\Big]+\Big[{n+2\over 6}\Big]+\Big[{n+4\over 6}\Big]=\Big[{n+2\over 2}\Big]+\Big[{n+3\over 6}\Big]$$
