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  "body": "\n## Statement\n\nLet $m$ and $n$ be natural numbers, and let $F_k$ be the [[Fibonacci numbers|0GA9K57A39]].\nThen\n$$F_{m+n+1} = F_m F_n + F_{m+1} F_{n+1}.$$\n\n## Remarks\n\nWith $m = n$ it gives $F_{2n+1} = F_n^2 + F_{n+1}^2$.\nAn equivalent form is $F_{m+n} = F_{m-1} F_n + F_m F_{n+1}$ for $m \\ge 1$.\nThe formula is the basis of fast algorithms that compute $F_n$ with $O(\\log n)$ arithmetic operations.\n\n## Explanations\n\n### L1 human\n\nTo get a Fibonacci number with a large index, you can combine Fibonacci numbers with smaller indices.\nFor example, with $m = 2$ and $n = 3$: $F_6 = F_2 F_3 + F_3 F_4 = 1 \\cdot 2 + 2 \\cdot 3 = 8$.\n",
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