This sequence appears to involve cyclic patterns of numbers, incrementing and wrapping around at each iteration. Here's a breakdown of the observations:

Pattern Analysis

1. Structure of the Sequence:

   • Numbers are grouped in rows indexed from 000 to 029.
   • The sequence cycles numbers from 0 to 9 repeatedly.

2. Row Cycles:

   • Each block of rows shifts numbers by two positions, wrapping around at 9.

   • Example:

     • Row 000-009: Starts at 4 and cycles through 4, 5, 6, 7, 8, 9, 0, 1, 2, 3.
     • Row 010-019: Starts at 6 and cycles through 6, 7, 8, 9, 0, 1, 2, 3, 4, 5.
     • Row 020-029: Starts at 8 and cycles through 8, 9, 0, 1, 2, 3, 4, 5, 6, 7.

3. Offset Rule:

   • Each row starts 2 numbers higher than the starting value of the previous row (modulo 10).

General Formula

For a number at row $r$ and column $c$, you can calculate the value as: $\text{Value}(r, c) = (S_r + c) \mod 10$ where:

• $S_r$ is the starting value of row $r$, calculated as $(4 + 2r) \mod 10$.
• $c$ is the column index (0 to 9).

Questions to Expand Understanding

1. Can you find the starting number for any given row using the general rule?
2. How would the pattern change if the offset per row was $+3$ instead of $+2$?
3. What are the implications of wrapping numbers in a modulo 10 system for real-world applications?
4. Could you generalize this sequence to other bases (e.g., modulo 12 or modulo 5)?
5. How can we encode this sequence efficiently using an algorithm?

Tip

When dealing with cyclic sequences, always check for recurring increments and how values wrap around at specific thresholds. Modulo arithmetic is a powerful tool for uncovering such patterns.