
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The positive integer  $n$  is given and for all positive integers  $k$ ,  $1\leq k\leq n$ , denote by  $a_{kn}$  the number of all ordered sequences  $(i_1,i_2,\ldots,i_k)$  of positive integers which verify the following two conditions: 

a)  $1\leq i_1<i_2< \cdots i_k \leq n$ ;

b)  $i_{r+1}-i_r \equiv 1 \pmod 2$ , for all  $r \in\{1,2,\ldots,k-1\}$ . 

Compute the number  $a(n) = \sum\limits_{k=1}^n a_{kn}$ . 

*Ioan Tomescu*

Present the answer in LaTex format: \boxed{Your answer}
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