### n-series

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**Sun Oct 30, 2016 7:55 pm**Given any integer $n\geq 3$. A finite series is called $n$-series if it satisfies the following two conditions

$1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$

$2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$

How many $n$-series are there $?$

$1)$ It has at least $3$ terms and each term of it belongs to $\{ 1,2,...,n\}$

$2)$ If series has $m$ terms $a_1,a_2,...,a_m$ then $(a_{k+1}-a_k)(a_{k+2}-a_k)<0$ for all $k=1,2,...,m-2$

How many $n$-series are there $?$