The numbers 3, 6, 4, 8, 6, 12, 10 form a deceptively simple pattern that alternates between two elementary operations: doubling the previous term and then subtracting two. This sequence, which we will refer to as the alternating doubling‑subtraction (ADS) sequence, provides a fertile ground for exploring recurrence relations, closed‑form expressions, and growth behavior. In what follows, we dissect the pattern, derive its mathematical description, and examine some of its intriguing properties Easy to understand, harder to ignore..
1. Pattern Recognition
At first glance the sequence appears irregular, yet a closer inspection reveals a strict rule:
- Start with 3.
- Double the current term → 3 × 2 = 6.
- Subtract 2 from the result → 6 − 2 = 4.
- Double the new term → 4 × 2 = 8.
- Subtract 2 → 8 − 2 = 6.
- Double → 6 × 2 = 12.
- Subtract 2 → 12 − 2 = 10.
The process repeats indefinitely, producing the infinite sequence
[ 3,;6,;4,;8,;6,;12,;10,;20,;18,;36,;34,;72,;70,;\dots ]
Each pair of consecutive terms can be seen as a cycle consisting of a multiplication step followed by a subtraction step.
