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  "body": "\n## Definition\n\nLet $X$ be a set and let $k$ be a positive integer.\nA recurrence relation of order $k$ on $X$ is an equation\n$$u_{n+k} = f(n, u_n, u_{n+1}, \\dots, u_{n+k-1}) \\qquad (n \\ge 0)$$\nfor a sequence $(u_n)_{n \\ge 0}$ in $X$, where $f : \\mathbb{N} \\times X^k \\to X$ is a given function.\n\nA solution is a sequence that satisfies the equation for all $n \\ge 0$.\n\n## Remarks\n\nFor each choice of initial values $u_0, \\dots, u_{k-1}$ there is exactly one solution. This follows by induction on $n$.\nExamples: the [[factorial|0WFCGVMRRC]] satisfies $u_{n+1} = (n + 1) u_n$ of order $1$. The [[Fibonacci sequence|0GA9K57A39]] satisfies $u_{n+2} = u_{n+1} + u_n$ of order $2$.\nAn important special case is a [[linear recurrence with constant coefficients|081THG13JZ]].\n\n## Explanations\n\n### L1 human\n\nA recurrence relation gives each term of a sequence from the terms just before it.\nOnce you know the first few terms, the rule determines all the others.\n",
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