Papers › Concatenations of Terms of an Arithmetic Progression
Concatenations of Terms of an Arithmetic Progression
Florian Luca, Bertrand Teguia Tabuguia
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Let (u(n))_(n∈ℕ) be an arithmetic progression of natural integers in base b∈ℕ∖{0,1}. We consider the following sequences: s(n)=u(0)u(1)⋯u(n) ᵇ formed by concatenating the first n+1 terms of (u(n))_(n∈ℕ) in base b from the right; s_g(n) = u(n)u(n-1)⋯u(0)ᵇ; and (s_*(n))_(n∈ℕ), given by s_*(0)=u(0), s_*(n)=s(n)s_g(n-1)ᵇ, n≥1. We construct explicit formulae 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 (s(n))_(n∈ℕ) and (s_g(n))_(n∈ℕ) in the decimal base when (u(n))_(n∈ℕ)=ℕ∖{0}.
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