Sequence of numbers

We've got a sequence of numbers {1,1}. We are creating a new sequence by adding to neighbouring elements. After first (n=1) adding we've sequence {1,2,1}, for n=2 we've a sequence {1,3,2,3,1} .... etc. Prove that sum of cubes of elements after n-step is equal $9*7^{n-1}+1$