Stern's diatomic sequence: s(0)=0, s(1)=1, s(2n)=s(n), s(2n+1)=s(n)+s(n+1). First terms: 0,1,1,2,1,3,2,3,1,4,3,5,2,5,3,4... The consecutive ratios s(n)/s(n+1) for n≥1 list every positive rational exactly once in lowest terms!
How to Use
Enter n:
- s(n): The n-th term
- Sequence: First n terms
- Rationals: s(n)/s(n+1)
Enumerating Rationals
s(1)/s(2)=1/1, s(2)/s(3)=1/2, s(3)/s(4)=2/1, s(4)/s(5)=1/3, s(5)/s(6)=3/2, ... This is the Calkin-Wilf enumeration. Every positive rational appears exactly once, already in lowest terms.
Binary Connection
s(n) counts the number of ways to write n in 'hyperbinary': representations using powers of 2 where each power is used at most twice. Also: s(n) = number of odd binomial coefficients in row n−1 of Pascal's triangle (mod 2 connection).
Step-by-Step Instructions
- 1Enter n.
- 2View s(n).
- 3See sequence.
- 4Explore rationals.
- 5Check binary.