Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions

Abstract

The odd and even symmetric Dellac numbers are the specializations $a_n=f_n(0)$ and $e_n=f_n(1)$ of the Randrianarivony--Zeng family $f_n(s)=2^{-n}{D}_n(s)$. Using a finite Stieltjes-fraction truncation, we establish the sharp congruence $v_3(a_n)=1$ for $n\ge 2$ and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer $s\ge 0$ and $k\ge 1$, we give and analyse formulas for the \emph{least} eventual period of $f_n(s)$ modulo $3^k$ and its \emph{least starting index}. The period is governed by five linear factors in $s$; the onset follows a nine-class formula modulo $9$. In particular, for $k\ge 3$ the even symmetric Dellac period is three times the odd period and begins one index earlier. The proofs combine a finite Stieltjes-fraction certificate, a graded $3$-adic pole expansion, and an explicit finite top-degree certificate over $\F_3$. Finite integer-arithmetic checks supplement the proofs. We also record the connection with the type-$B$ homogenized Linial arrangement.

arXiv comment19 pages