Having a finite sequence of numbers given, we create a new sequence by inputting in each step between every pair of two adjacent numbers a new number equal to their sum. We start with (1,1), in the second step we have (1,2,1), in the third (1,3,2,3,1) etc. For every

calculate the sum of the cubes of the numbers being part of the sequence acquired in the nth step.

What we know is that in every step, for a sequence of lenght n we'll get n-1 new numbers being the sums of the adjacents so the next sequence will be 2n-1. The sum of the first is

, then we have

, then the_sum_so_far+

. The useful property is that the sequence is symmetrical having some k pairs of numbers on both sides of the central 2 and always has an odd amount of numbers - only the first step is even. How does that lead to a solution, though? Could you please help?