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  "body": "\n## Definition\n\nLet $R$ be a commutative ring, and let $E$ be a [[linear recurrence with constant coefficients|081THG13JZ]] over $R$, of order $d$ with coefficients $c_0, \\dots, c_{d-1}$.\nA solution of $E$ is a sequence $u : \\mathbb{N} \\to R$ such that\n$$u_{n+d} = \\sum_{i=0}^{d-1} c_i\\, u_{n+i} \\quad \\text{for all } n \\in \\mathbb{N}.$$\n\n## Remarks\n\nEach choice of the initial values $u_0, \\dots, u_{d-1}$ gives exactly one solution.\nThe sum of two solutions is a solution, and a multiple of a solution is a solution. So the solutions form the [[space of solutions|05BDYVQQ0T]].\nFor $u_{n+2} = u_{n+1} + u_n$, the Lucas numbers $2, 1, 3, 4, 7, 11, \\dots$ form a solution.\n\n## Explanations\n\n### L1 human\n\nA solution is a sequence that obeys the rule of the recurrence at every step.\nThe rule alone does not fix the sequence: you also choose the first $d$ terms.\n",
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