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  "body": "\n## Statement\n\nLet $n$ be a natural number, and let $F_m$ be the [[Fibonacci numbers|0GA9K57A39]].\nThen\n$$F_{n+1} = \\sum_{k=0}^{\\lfloor n/2 \\rfloor} \\binom{n - k}{k}.$$\n\n## Remarks\n\nExample: $F_6 = \\binom{5}{0} + \\binom{4}{1} + \\binom{3}{2} = 1 + 4 + 3 = 8$.\nCombinatorial proof: $F_{n+1}$ counts the tilings of a strip of length $n$ by squares and dominoes. A tiling with $k$ dominoes has $n - k$ tiles, and $\\binom{n-k}{k}$ choices fix the positions of the dominoes.\nIn [[Pascal's triangle|05DME4TZ3K]], the sum runs along a shallow diagonal.\n\n## Explanations\n\n### L1 human\n\nAdd the numbers of Pascal's triangle along the shallow diagonals, and you get the Fibonacci numbers.\nEach term counts the ways to tile a strip with a fixed number of dominoes.\n",
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