Farey Sequence Generator

How it is calculated

The tool starts from the first two neighbours 0/1 and 1/n and produces each next fraction from the previous two with the neighbour recurrence: k = floor((n + b) / d), then next = (k·c − a) / (k·d − b), where a/b and c/d are the last two fractions. It stops when it reaches 1/1. Because of this recurrence every fraction comes out already in lowest terms and in increasing order, with no sorting or gcd step. The order n is rounded to the nearest whole number and must be between 1 and 50; anything outside that range gives no result. The count it reports is the number of fractions, which equals 1 + φ(1) + φ(2) + … + φ(n), where φ is Euler's totient.

Common mistakes

• Expecting fractions that are not in lowest terms. 2/4 or 3/6 never appear because 1/2 already covers them, so the count is smaller than all pairs a/b with b ≤ n.
• Confusing the order n with the number of terms. F_5 has 11 terms, not 5; the count grows roughly with n².
• Entering a decimal such as 4.6. The tool rounds it to 5, so you get F_5.
• Entering an order above 50 and expecting a list. The tool only handles 1 to 50 and returns nothing beyond that.
• Forgetting that both 0/1 and 1/1 are included, which matters when you compare the count with a totient sum.