Van Eck's sequence (A181391): start with 0. At each step: if current value is new, write 0. If seen before, write the gap since its last occurrence. Result: 0,0,1,0,2,0,2,2,1,6,0,5,0,2,... Deceptively simple rule, incredibly complex behavior.
How to Use
Enter terms count:
- Sequence: Generated terms
- New/Seen: First occurrence?
- Zeros: How many 0s
The Zeros
Every time a new number first appears, the next term is 0 (it was new). So 0 appears extremely often — conjecture: 0 appears with density 0 but infinitely often. The sequence returns to 0 again and again.
Open Questions
- Does every positive integer eventually appear?
- What is the density of 0s?
- Is there a pattern in the growth rate?
- All unknown! Despite the simple rule.
Step-by-Step Instructions
- 1Enter terms.
- 2Generate sequence.
- 3Track occurrences.
- 4Note zeros.
- 5Analyze patterns.