1,1,1,3,5,9...

This reminds us of the Fibonacci sequence.

1,1,2,3,5,8,13,...

Rule for the Fibonacci sequence:  Start with two 1's.
Get the 3rd term and every other term by adding the 
two preceding terms.

So go one step further:

Rule for the this sequence:  Start with three 1's.
Get the 4th term and every other term by adding the
three preceding terms.

1,1,1,3,5,9...
1+1+1=3
  1+1+3=5
    1+3+5=9
      3+5+9=?
        5+9+?=?
          9+?+?=?
            ?+?+?=?
               etc. 

Put numbers where those question marks are.
You extend that pattern as far as you like.

Edwin