二項係数の和とフィボナッチ

$\sum_{i} (i N - i) = F_{N}$

where $F_{0} = F_{1} = 1, F_{n} = F_{n - 2} + F_{n - 1}$

$F_{N} = \sum_{i \geq 0} (i N - i) = 1 + \sum_{i \geq 1} (i N - i)$ $= 1 + \sum_{i \geq 1} ((i N - i - 1) + (i - 1 N - i - 1))$ $= 1 + \sum_{i \geq 1} (i N - i - 1) + \sum_{i \geq 1} (i - 1 N - i - 1)$ $= 1 + \sum_{i \geq 1} (i N - i - 1) + \sum_{j \geq 0} (j N - j - 2)$ $= \sum_{i \geq 0} (i N - i - 1) + \sum_{j \geq 0} (j N - j - 2)$ $= \sum_{i \geq 0} (i ( N - 1 ) - i) + \sum_{i \geq 0} (i ( N - 2 ) - i)$ $= F_{N - 1} + F_{N - 2}$

二項係数とフィボナッチ数列

🪴 Quartz 4.0

二項係数の和とフィボナッチ

Graph View

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