How to get the sum of unit steps and unit ramps from a discrete signal?
To avoid mistakes, let us first divide the original sequences by $2$ (for the sake of linearity):

0 0 1 2 3 3 3 0 0 0 0$\ldots$

It can be interpreted as a piecewise linear sequence, so you can expect a solution with about  three degrees of freedom with ramps and steps. As these basic primitives are causal, you can use the method of deflation (or successive approximations). This is a quite common approach, used for instance in subspace decomposition, or in basis pursuit in signal processing. The idea is: find a first approximation that matches what you want approximately, subtract it from the data, and re-iterate the processing. The convergence is not always guaranteed, but it can give you insights or heuristics  for a deeper approach. Here, it is likely to work fine.

You can start from left to right, build the obvious, remove it from the data, and iterate the process.

The first 0 0 1 2 3 can be approximated with ramp $r(n-1)$. Subtract  $r(n-1)$ from the data, and get: 

0 0 0 0 0 1 2 6 7 8 9$\ldots$

You can compensate the "novel" left-most positive  slope 0 1 2 with $-r(n-4)$. You now get:

0 0 0 0 0 0 0 3 3 3 3$\ldots$

and then the $-3u(n-7)$ finishes the job. Oh no, don't forget to multiply by two again.

Here, the exercise is relatively easy to manage. Such a morphological decomposition can be used on more complicated signals. If you want more details, you can check for instance Automatic decomposition of time series into step, ramp, and impulse primitives, Galati &amp; Simaan, 2006:


  Time series data that can be modeled as linear combinations of
  weighted and shifted primitive functions such as ramps, steps and
  impulses are representative of many industrial, manufacturing, and
  business processes. Data of this type also are found in statistical
  process control, structural health monitoring, and other system
  diagnosis applications. Often, the existence of one or more of these
  primitive functions may be indicative of the occurrence of a specific
  process event, making their detection and interpretation of great
  interest. The human eye is an exceptional tool at this kind of pattern
  recognition. However, for processes that generate large amounts of
  data the human eye encounters difficulties related to speed and
  consistency necessitating an automated approach.