Explicit formulas for concatenations of arithmetic progressions

arXiv2201.07127MaRDI QIDQ6388571

Bertrand Teguia Tabuguia, Florian Luca

Publication date: 14 January 2022

Abstract: The sequence $(Sm(n){)}_{ngeqslant0}$: $1$, $12$, $123$, $ldots$ 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 $(Sm(n){)}_{ngeqslant0}$ or the concatenation of odd integers: $1$, $13$, $135$, $ldots$; the left-concatenation like the reverse of Smarandache consecutive numbers $(Smr(n){)}_{ngeqslant0}$: $1$, $21$, $321$, $ldots$; and the concatenation of right-concatenation and left-concatenation like $1$, $121$, $12321$, $1234321$,$ldots$ formed by $Sm(n)$ and $Smr(n-1)$ for $ngeqslant1$, 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 $(Sm(n){)}_{ngeqslant0}$ and $(Smr(n){)}_{ngeqslant0}$.

Has companion code repository: https://github.com/t3gu1a/concatenations

Mathematics Subject Classification ID

Symbolic computation and algebraic computation (68W30) Software, source code, etc. for problems pertaining to number theory (11-04) Special sequences (11K31) Calculation of integer sequences (11Y55)

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