Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions
Abstract
The odd and even symmetric Dellac numbers are the specializations an=fn(0) and en=fn(1) of the Randrianarivony--Zeng family fn(s)=2−nDn(s). Using a finite Stieltjes-fraction truncation, we establish the sharp congruence v3(an)=1 for n≥2 and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer s≥0 and k≥1, we give and analyse formulas for the \emph{least} eventual period of fn(s) modulo 3k 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≥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 F3. Finite integer-arithmetic checks supplement the proofs. We also record the connection with the type-B homogenized Linial arrangement.
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BibTeX
@misc{newton2026exact,
title = {{Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions}},
author = {Tony Newton and Jiang Zeng},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05962},
eprint = {2611.05962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05962},
}
RIS
TY - UNPB
TI - Exact $3$-Adic Periods of Symmetric Dellac Numbers and Their Polynomial Extensions
AU - Tony Newton
AU - Jiang Zeng
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05962
UR - https://arxiv.org/abs/2611.05962
AB - 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^−nD_n(s). Using a finite Stieltjes-fraction truncation, we establish the sharp congruence v_3(a_n)=1 for n≥2 and obtain explicit congruences modulo $81$ for both sequences. We also establish universal eventual periodicity modulo arbitrary positive integers. For every integer s≥0 and k≥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≥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.
ER -