There exist several characterizations of k-regular sequences, all of which are equivalent. Some common characterizations are as follows. For each, we take R′ to be a See more

Ruler sequence
Let $${\displaystyle s(n)=\nu _{2}(n+1)}$$ be the $${\displaystyle 2}$$-adic valuation of $${\displaystyle n+1}$$. The ruler sequence $${\displaystyle s(n)_{n\geq 0}=0,1,0,2,0,1,0,3,\dots }$$ (OEIS: A007814 See more

The notion of k-regular sequences was first investigated in a pair of papers by Allouche and Shallit. Prior to this, Berstel and Reutenauer studied the theory of rational series, … See more

Given a candidate sequence $${\displaystyle s=s(n)_{n\geq 0}}$$ that is not known to be k-regular, k-regularity can typically be proved directly from the definition by calculating elements of the kernel of $${\displaystyle s}$$ and proving that all elements of … See more