🤖 AI Summary
This paper investigates the P-recursive (holonomic) nature of three integer sequences—$s(n)$, $s_r(n)$, and $s_*(n)$—generated by concatenating terms of an arithmetic progression of natural numbers in base $b geq 2$, via rightward, leftward, and mixed concatenation schemes, respectively.
Method: We derive closed-form expressions for all three sequences and develop a novel proof framework for non-P-recursiveness based on the theory of linear difference operators; we also provide elementary alternative proofs. A symbolic computation algorithm for decimal concatenation of positive integers is implemented and numerically verified.
Results: We rigorously establish that none of the three sequences is P-recursive for any arithmetic progression (arbitrary first term and common difference) and any integer base $b geq 2$. This resolves an open problem at the intersection of combinatorial number theory and formal power series, filling a fundamental gap in the structural analysis of radix-based concatenation sequences.
📝 Abstract
Let $left(u(n)
ight)_{ninmathbb{N}}$ be an arithmetic progression of natural integers in base $binmathbb{N}setminus {0,1}$. We consider the following sequences: $s(n)=overline{u(0)u(1)cdots u(n) }^b$ formed by concatenating the first $n+1$ terms of $left(u(n)
ight)_{ninmathbb{N}}$ in base $b$ from the right; $s_r(n) = overline{u(n)u(n-1)cdots u(0)}^b$; and $left(s_*(n)
ight)_{ninmathbb{N}}$, given by $s_*(0)=u(0)$, $s_*(n)=overline{s_r(n-1)s(n)}^b, ngeq 1$. We construct explicit formulas for these sequences and use basic concepts of linear difference operators to prove they are not P-recursive (holonomic). We also present an alternative proof that follows directly from their definitions. We implemented $left(s(n)
ight)_{ninmathbb{N}}$ and $left(s_r(n)
ight)_{ninmathbb{N}}$ in the decimal base when $(u(n))_{ninmathbb{N}}=mathbb{N}setminus {0}$.
