Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One
Abstract
This work presents a tabular method for constructing quasi-figurate numbers. The method is based on iteratively adding a chosen figurate series (starting from zero) to the natural series (starting from one). If the triangular series is used as the offset, the iteration generates all classical figurate numbers. If a different figurate series is used (for example, quadrilateral numbers), new families of numbers arise — quasi-figurate numbers.
The first part of the work describes the method, constructs tables of levels t = 1, ..., 12, and derives horizontal and vertical formulas for the tables. The formulas are verified numerically (1500 values, no discrepancies). Python code for constructing the tables and verifying the formulas is provided.
Among the quasi-figurate numbers, several remarkable sequences appear naturally:
- Mersenne numbers (1, 3, 7, 15, 31, ...) arise as the first-level quasi-triangular numbers (row k = 3, level t = 1).
- Repunits (1, 11, 111, 1111, ...) arise in the table of quasi-eleven-gonal numbers (m = 11), row k = 10.
- The number 123456789 and its continued pattern (1, 12, 123, 1234, ...) arise in the table of quasi-twelve-gonal numbers (m = 12), row k = 10.
- The number 98765432 and its continued pattern (1, 10, 99, 988, ...) arise in the table of quasi-ten-gonal numbers (m = 10), row k = 10.
- The number 8641975 and its continued pattern (1, 9, 87, 865, ...) arise in the table of quasi-nine-gonal numbers (m = 9), row k = 10.
These structures arise naturally from the general formula — they are not introduced by hand.
An interesting observation is also described: applying the formula for small k (k = 1, 2) yields three types of rows — negative mirrors, the natural series, and Mersenne numbers. The formula for k = 1 gives "Mersenne in the mirror" (Mersenne numbers with correction -2n, reflected into the negative region).
Possible areas of application (noted as expectations, not proven facts) include fractals, automatic sequences, quasicrystals, modular arithmetic, dynamical systems, cryptography, information theory, and music.
Related
- Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two
- Monochromatic symmetric subsets in finite colourings of infinite abelian groups
- Hodge--Tjurina Numbers, Tjurina Spectra, and Thom--Sebastiani Formulas
- Mirny's Cosmodynamics: Notes. Note No. 3 Rational loops — subsets: why circle searchers don't find them by construction
- Separability of Subsets in Infinite Groups
BibTeX
@misc{helmdach2026infinite,
title = {{Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One}},
author = {Emma Helmdach},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23537119},
url = {https://zenodo.org/records/23537119},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One
AU - Emma Helmdach
PY - 2026
DA - 2026/10/21
PB - Zenodo
DO - 10.5281/zenodo.23537119
UR - https://zenodo.org/records/23537119
AB - This work presents a tabular method for constructing quasi-figurate numbers. The method is based on iteratively adding a chosen figurate series (starting from zero) to the natural series (starting from one). If the triangular series is used as the offset, the iteration generates all classical figurate numbers. If a different figurate series is used (for example, quadrilateral numbers), new families of numbers arise — quasi-figurate numbers. The first part of the work describes the method, constructs tables of levels t = 1, ..., 12, and derives horizontal and vertical formulas for the tables. The formulas are verified numerically (1500 values, no discrepancies). Python code for constructing the tables and verifying the formulas is provided. Among the quasi-figurate numbers, several remarkable sequences appear naturally: - Mersenne numbers (1, 3, 7, 15, 31, ...) arise as the first-level quasi-triangular numbers (row k = 3, level t = 1). - Repunits (1, 11, 111, 1111, ...) arise in the table of quasi-eleven-gonal numbers (m = 11), row k = 10. - The number 123456789 and its continued pattern (1, 12, 123, 1234, ...) arise in the table of quasi-twelve-gonal numbers (m = 12), row k = 10. - The number 98765432 and its continued pattern (1, 10, 99, 988, ...) arise in the table of quasi-ten-gonal numbers (m = 10), row k = 10. - The number 8641975 and its continued pattern (1, 9, 87, 865, ...) arise in the table of quasi-nine-gonal numbers (m = 9), row k = 10. These structures arise naturally from the general formula — they are not introduced by hand. An interesting observation is also described: applying the formula for small k (k = 1, 2) yields three types of rows — negative mirrors, the natural series, and Mersenne numbers. The formula for k = 1 gives "Mersenne in the mirror" (Mersenne numbers with correction -2n, reflected into the negative region). Possible areas of application (noted as expectations, not proven facts) include fractals, automatic sequences, quasicrystals, modular arithmetic, dynamical systems, cryptography, information theory, and music.
ER -