I recently posed a question on Code Golf on what I'm terming "Fibonacci Digits", where each number is the sum of the previous two digits, not necessarily the previous two numbers. So, for instance, the series beginning $1,1$ is $1,1,2,3,5,8,13,4,7,11,2\dots$ and cycles from there.
As I note in the various examples I bring there, unless something in the series manages to add to either $0$ or $14$, the series seems to always loop to some rotation of $2,3,5,8,13,4,7,11$. Further, it always reaches $0$ or $14$ no later than the third entry in the series. For instance, $S_1=7,7,14,5,9\dots$, $S_2=14,5,9\dots$, $S_3=68,14,5,9\dots$.
In other words, it seems that for $0\le(a,b)\le9$, then, for $a+b=c$:
- If $c<10$, let $d=b+c$
- If $c\ge10$, $c=10x+y$ for some integers $0\le(x,y)\le9$. Let $d=x+y$.
- Repeat the above using $c$ and $d$.
Eventually, assuming that you don't start with $0$, this algorithm will yield either $11$ or $14$. Seemingly.
My question is two parts:
- Is my assessment correct?
- If so, why?