Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two
Abstract
This paper is the second part of a study on quasi-figurate numbers. The first part, "Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One", is available at https://doi.org/10.5281/zenodo.23537119.
This paper investigates the periodicity of quasi-figurate numbers — infinite families of numerical sequences obtained by iteratively adding a chosen figurate series (written from zero) to the natural numbers — for large indices n.
For k = 10 (the base of the decimal system), the decimal notation of f_n(10) contains a repeating block of 9 digits determined by the fraction (m−2) / 81, where m is the gonality of the offset. A summary table of blocks for m = 3, …, 13 is given. For m = 9q + 2, the block is a repunit (a number consisting of repeated 1s, 2s, 3s, …), confirmed numerically for m = 11, 20, 29, 38.
For k ≠ 10, no block is visible in decimal notation, but periodicity does not disappear: it passes into the ratio f_(n+k−1)(k) / f_n(k), which tends to k^(k−1) as n → ∞. This is confirmed numerically for k = 3, …, 15 (at n = 100), with the difference decreasing exponentially.
All observations are obtained numerically; Python/SageMath code for reproducing the tables and checks is included. The observations are stated as numerical results, not as theorems.
Related
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BibTeX
@misc{helmdach2026infinite,
title = {{Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two}},
author = {Emma Helmdach},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23555682},
url = {https://zenodo.org/records/23555682},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two
AU - Emma Helmdach
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23555682
UR - https://zenodo.org/records/23555682
AB - This paper is the second part of a study on quasi-figurate numbers. The first part, "Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One", is available at https://doi.org/10.5281/zenodo.23537119. This paper investigates the periodicity of quasi-figurate numbers — infinite families of numerical sequences obtained by iteratively adding a chosen figurate series (written from zero) to the natural numbers — for large indices n. For k = 10 (the base of the decimal system), the decimal notation of f_n(10) contains a repeating block of 9 digits determined by the fraction (m−2) / 81, where m is the gonality of the offset. A summary table of blocks for m = 3, …, 13 is given. For m = 9q + 2, the block is a repunit (a number consisting of repeated 1s, 2s, 3s, …), confirmed numerically for m = 11, 20, 29, 38. For k ≠ 10, no block is visible in decimal notation, but periodicity does not disappear: it passes into the ratio f_(n+k−1)(k) / f_n(k), which tends to k^(k−1) as n → ∞. This is confirmed numerically for k = 3, …, 15 (at n = 100), with the difference decreasing exponentially. All observations are obtained numerically; Python/SageMath code for reproducing the tables and checks is included. The observations are stated as numerical results, not as theorems.
ER -