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In mathematics, the Farey sequence of order n is the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.

F₁ = { 01_, 11 }

F₂ = { 01_, 12_, 11 }

F₃ = { 01_, 13_, 12_, 23_, 11 }

F₄ = { 01_, 14_, 13_, 12_, 23_, 34_, 11 }

F₅ = { 01_, 15_, 14_, 13_, 25_, 12_, 35_, 23_, 34_, 45_, 11 }

F₆ = { 01_, 16_, 15_, 14_, 13_, 25_, 12_, 35_, 23_, 34_, 45_, 56_, 11 }

F₇ = { 01_, 17_, 16_, 15_, 14_, 27_, 13_, 25_, 37_, 12_, 47_, 35_, 23_, 57_, 34_, 45_, 56_, 67_, 11 }

F₈ = { 01_, 18_, 17_, 16_, 15_, 14_, 27_, 13_, 38_, 25_, 37_, 12_, 47_, 35_, 58_, 23_, 57_, 34_, 45_, 56_, 67_, 78_, 11 }