What is the Number Pattern Pattern?

The phrase "number pattern pattern" is a bit unclear, but we can interpret it to mean either "what constitutes a number pattern" or "patterns of number patterns." Let's address both possible interpretations.

What Constitutes a Number Pattern?

A number pattern is a sequence of numbers that follows a specific rule or set of rules. This rule determines how each subsequent number in the sequence is generated. The reference provides several examples:

• Even Numbers Pattern: 2, 4, 6, 8, 10, 12, 14, 16, 18, … (Each number is 2 greater than the previous number, or multiples of 2).
• Odd Numbers Pattern: 3, 5, 7, 9, 11, 13, 15, 17, 19, … (Each number is 2 greater than the previous number, starting with 3).
• Fibonacci Numbers Pattern: 1, 1, 2, 3, 5, 8, 13, 21, … (Each number is the sum of the two preceding numbers).

Patterns of Number Patterns

This interpretation is more abstract. It refers to the idea that the rules that generate number patterns can themselves exhibit patterns. Here are a few (more complex) examples:

• Arithmetic Progressions with Increasing Differences: Consider the sequence of patterns:

  • Pattern 1: 1, 2, 3, 4... (Difference of 1)
  • Pattern 2: 1, 3, 5, 7... (Difference of 2)
  • Pattern 3: 1, 4, 7, 10... (Difference of 3)
  • Pattern 4: 1, 5, 9, 13... (Difference of 4)
    The differences between consecutive terms are themselves increasing arithmetically (1, 2, 3, 4...).

• Combining Patterns: A pattern might involve alternating between two different rules. For example:

  • Generate an even number, then a Fibonacci number, then an even number, etc.

• Patterns Based on Prime Numbers: Create a sequence where the difference between terms is determined by prime numbers. For example:

  • Start with 1:
    • Next term: 1 + 2 = 3
    • Next term: 3 + 3 = 6
    • Next term: 6 + 5 = 11
    • Next term: 11 + 7 = 18
    • The sequence is 1, 3, 6, 11, 18...

In essence, the "number pattern pattern" signifies a meta-level analysis where the underlying rules governing numerical sequences themselves form a pattern or exhibit predictable behavior.