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  "body": "\n## Definition\n\nLet $n$ be a positive integer.\nThe Farey sequence $F_n$ of order $n$ is the increasing sequence of all fractions $\\frac{a}{b}$ with $0 \\le a \\le b \\le n$ and $\\gcd(a, b) = 1$.\nEach fraction is in lowest terms, and $F_n$ starts with $\\frac{0}{1}$ and ends with $\\frac{1}{1}$.\n\n## Remarks\n\nFor example, $F_5 = \\left( \\frac{0}{1}, \\frac{1}{5}, \\frac{1}{4}, \\frac{1}{3}, \\frac{2}{5}, \\frac{1}{2}, \\frac{3}{5}, \\frac{2}{3}, \\frac{3}{4}, \\frac{4}{5}, \\frac{1}{1} \\right)$.\nIf $\\frac{a}{b} \u003c \\frac{c}{d}$ are neighbours in $F_n$, then $bc - ad = 1$.\nThe length of $F_n$ is $1 + \\sum_{k=1}^{n} \\varphi(k)$, with $\\varphi$ [[Euler's totient function|0VV04M41AN]].\nThe sequences are named after John Farey, whose letter about them appeared in 1816.\n\n## Explanations\n\n### L1 human\n\nWrite all fractions between $0$ and $1$ with denominators up to $n$, in lowest terms and in increasing order.\nNeighbouring fractions in this list are always very close in a precise sense: their cross difference is $1$.\n",
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