I am writing up an assignment with includes many identities of Fibonacci numbers. I have made up the following notation (here $f_n$ is the number of tilings of an $n$-board by dominoes and squares - a well known interpretation of Fibonacci numbers):
Given an $n$-board, and some sequence $seq$ of dominos and squares, let $f_n^{seq}$ represent all tilings of an $n$-board that end with the sequence $seq$. So for example $f_n^{DS}$ are all tilings of an $n$-board that end with a domino succeeded by a square. Furthermore, for repeating sequences write: $ \underbrace{seq,seq, \cdots, seq}_{n-\text{times}} = n\cdot seq $ So for example $f_n^{SSDSDSDSDS} = f_n^{2S4DS}$. Let the size of a sequence $|seq|$ be the number of tiles the sequence occupies, so $|SS4DS| = 14$. It is clear that: $ f_n^{seq} = f_{n - |seq|} $
How may this be improved? Are statements such as
$ f_{3n-3k} = f_{3n}^{kDS} = f_{3n}^{SSkDS} + f_{3n}^{DkDS} + f_{3n}^{(k+1)DS} = f_{3n-3k}^{SS} + f_{3n-3k}^{D} + f_{3n-3k}^{DS} = $ $ = 2f_{3n-3k-2} + f_{3n-3k-3} $
a bit understandable without explanation? (I do offer an explanation of course).