Self-Similar Structure in Quasi-10-gonal Numbers: The Case of Column 50

This note illustrates the possibilities of the quasi-figurate number method on a single example: quasi-10-gonal numbers, column 50. The example was chosen at random; we do not claim that it is special or unique. The goal is to show that even on one example non-trivial structures are visible. Limiting values for column 50 are reached only after 55 iterations (for rows 1..3005); stabilization was verified numerically (values do not change when passing from 55 to 56, 57, ..., 60 iterations). We investigate exactly these limiting values, not the intermediate results of 15 iterations. The main observations: (1) adjacent rows (starting from row 3) form an arithmetic progression with difference equal to row 2; (2) for each row r >= 3, paired rows are determined by the step d_k = (9r - 18) * 10^(k-1); (3) for all verified rows r = 3..8 and k = 1, 2, 3 the general formula holds: row r + d_k = row r * 10^k - 50 * (10^k - 1); (4) for row 2 (offset) the structure is different: row 2 + 10^k = row 2 * 10^k + 50; (5) the observed structure is a structural self-similarity with scale 10 -- not periodicity, not a fractal, not a linear function. In addition, we studied the structure of columns 7, 11, and 50 for quasi-10-gonal numbers (limiting values after 55 iterations, rows 1..300) and observed a general pattern. Each column number c corresponds to a divisor c by which the numbers of the column are divided with the same period c into an integer. The remaining numbers are divided with a remainder, but the remainders have period c - 1 and always repeat after the next division by c into an integer. We wrote Python code for columns 7, 11, and 50 and determined the periods of remainders: for column 7 the period of remainders is 6 (3, 6, 2, 5, 1, 4), for column 11 it is 10 (8, 5, 2, 10, 7, 4, 1, 9, 6, 3), and for column 50 it is 49. We conjecture that this rule extends to all columns of all quasi-figurate numbers, but we do not claim it as a theorem: within a single work this cannot be verified. This opens the possibility for further research. The note includes Python code for constructing column 50 (55 iterations), for checking the paired rows, and for columns 7, 11, and 50. Numerical verification was performed with SageMath and DeepSeek algorithms. All ideas of the author are original. This note continues the first part (Infinite Subsets of Quasi-Figurate Numbers: Tables and Formulas for Construction. Part One, https://doi.org/10.5281/zenodo.23537119) and the second part (Infinite Subsets of Quasi-Figurate Numbers: Periodicity in Decimal Notation. Repunits. Part Two, https://doi.org/10.5281/zenodo.23555682).