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  "body": "\n## Example\n\nHofstadter sequences are integer sequences defined by nested [[nonlinear recurrences|7C7FBZGERM]], in which earlier values decide which earlier terms are used.\n\n1. The *Hofstadter $Q$-sequence* is given by $Q(1) = Q(2) = 1$ and $Q(n) = Q(n - Q(n-1)) + Q(n - Q(n-2))$ for $n \u003e 2$. It begins $1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, \\dots$.\n2. The *Hofstadter $G$-sequence* is given by $G(0) = 0$ and $G(n) = n - G(G(n-1))$ for $n \\ge 1$. It begins $0, 1, 1, 2, 3, 3, 4, 4, 5, \\dots$.\n\nCheck for $Q$: $Q(3) = Q(2) + Q(2) = 2$, $Q(4) = Q(2) + Q(3) = 3$, $Q(5) = Q(2) + Q(3) = 3$, $Q(6) = Q(3) + Q(3) = 4$.\nCheck for $G$: $G(1) = 1 - G(0) = 1$, $G(2) = 2 - G(1) = 1$, $G(3) = 3 - G(1) = 2$, $G(4) = 4 - G(2) = 3$.\nThe $G$-sequence has the closed form $G(n) = \\lfloor (n+1)/\\varphi \\rfloor$, where $\\varphi$ is the [[golden ratio|0SC8T7548D]].\nThe $Q$-sequence behaves erratically, and it is not known whether it is defined for all $n$, that is, whether $n - Q(n-1) \\ge 1$ and $n - Q(n-2) \\ge 1$ always hold.\n\n## Explanations\n\n### L1 human\n\nThese sequences come from Douglas Hofstadter's book *Gödel, Escher, Bach*.\nIn an ordinary recurrence, you always look back a fixed number of steps.\nHere, the values of the sequence tell you how far back to look, which makes the rule self-referential.\nSome of these sequences turn out to be tame, like $G$, while others, like $Q$, look chaotic and are still not understood.\n",
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