how this series is generated
1,1,1 2,1 2 3,1 2 3 4 5, 1 2 3 4 5 6 7 8
The last digit in the $\displaystyle n$-th block is the sum of the last digits in the two preceding blocks, with the initial condition that the first two terms are $\displaystyle 1$ and $\displaystyle 1$. This makes the $\displaystyle n$-th block the numbers $\displaystyle 1$ to $\displaystyle F_n$ where $\displaystyle F_n$ is the $\displaystyle n$-th Fibonacci number.
CB