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  "body": "\n## Summary\n\nA linear recurrence expresses a term of a sequence as a linear combination of earlier terms.\nIt may also add an extra term that depends only on the index.\nIt is homogeneous when this extra term is zero, and inhomogeneous otherwise.\nThe solutions of a homogeneous recurrence form a vector space.\nThe solutions of an inhomogeneous one form a translate of that space.\nSo solving an inhomogeneous recurrence reduces to finding one particular solution and the solutions of the homogeneous one.\n\n## Explanations\n\n### L1 human\n\nThe rule $u_{n+2} = u_{n+1} + u_n$ only mixes earlier terms, so it is homogeneous.\nThe rule $u_{n+1} = 2 u_n + 1$ adds a fixed amount at each step, so it is inhomogeneous.\nFor a homogeneous rule, the all-zero sequence is always a solution, and sums and multiples of solutions are again solutions.\nFor an inhomogeneous rule this fails: the zero sequence does not satisfy $u_{n+1} = 2 u_n + 1$.\nIts solutions are $u_n = c\\, 2^n - 1$: one particular solution $-1$, plus any solution $c\\, 2^n$ of $u_{n+1} = 2 u_n$.\n",
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