Explicit formulas for concatenations of arithmetic progressions

The sequence $(Sm(n))_{n\geqslant 0}$: $1$, $12$, $123$, $\ldots$ formed by concatenating the first $n+1$ positive integers is often called Smarandache consecutive numbers. We consider the more general case of concatenating arithmetic progressions and establish formulas to compute them. Three types of concatenation are taken into account: the right-concatenation like $(Sm(n))_{n\geqslant0}$ or the concatenation of odd integers: $1$, $13$, $135$, $\ldots$; the left-concatenation like the reverse of Smarandache consecutive numbers $(Smr(n))_{n\geqslant 0}$: $1$, $21$, $321$, $\ldots$; and the concatenation of right-concatenation and left-concatenation like $1$, $121$, $12321$, $1234321$,$\ldots$ formed by $Sm(n)$ and $Smr(n-1)$ for $n\geqslant1$, with the initial term $Sm(0)$. The resulting formulas enable fast computations of asymptotic terms of these sequences. In particular, we use our implementation in the Computer Algebra System Maple to compute billionth terms of $(Sm(n))_{n\geqslant0}$ and $(Smr(n))_{n\geqslant0}$.