AN EXPLICIT FORMULA FOR THE TERMS OF REGULAR PAPER-FLOOR SEQUENCES

Source abstract

In this paper, we derive an explicit formula for the $k$ th term of the regular paper-floor sequence $F_{P^{n}}$. For $1 \leq k \leq 2^{n}$, we introduce three finite auxiliary sequences that record the binary representation of $2^{n}-k$, the changes between consecutive augmented digits, and the > corresponding cumulative numbers of changes. Using these sequences, we express $\left(F_{P^{n}}\right)_{k}$ directly as a finite sum, without iterative reduction.