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  "body": "\n## Definition\n\nLet $R$ be a commutative ring and let $d$ be a natural number.\nA linear recurrence of order $d$ with constant coefficients over $R$ is given by coefficients $c_0, c_1, \\dots, c_{d-1} \\in R$.\nIt is the [[recurrence relation|07TZV87Y0T]]\n$$u_{n+d} = c_0 u_n + c_1 u_{n+1} + \\dots + c_{d-1} u_{n+d-1} \\qquad (n \\ge 0)$$\nfor sequences $(u_n)_{n \\ge 0}$ in $R$.\n\n## Remarks\n\nThis form is homogeneous. A non-homogeneous recurrence adds a term $g(n)$ that does not depend on $u$.\nThe [[Fibonacci sequence|0GA9K57A39]] satisfies the recurrence of order $2$ with $c_0 = c_1 = 1$.\nThe sequences that satisfy it are its [[solutions|0ZV2YKKZ58]]. Its [[characteristic polynomial|0BWDJGQYST]] is $X^d - c_{d-1} X^{d-1} - \\dots - c_1 X - c_0$.\n\n## Explanations\n\n### L1 human\n\nIn a linear recurrence with constant coefficients, each new term is a fixed weighted sum of the previous $d$ terms.\nThe weights do not change with $n$.\n",
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