Fibonacci Number as Sum of Binomial Coefficients/Mistake
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Source Work
1997: David Wells: Curious and Interesting Numbers:
- The Dictionary
- $5$
First Edition
- Lucas discovered a relationship between Fibonacci numbers and the binomial coefficients:
- $F_{n + 1} = \paren {\dfrac n 0} + \paren {\dfrac {n - 1} 1} + \paren {\dfrac {n - 2} 2} + \cdots$
- For example:
\(\ds F_{12} = 144\) | \(=\) | \(\ds \paren {\frac {11} 0} + \paren {\frac {10} 1} + \paren {\frac 9 2} + \paren {\frac 8 3} + \paren {\frac 7 4} + \paren {\frac 6 5}\) | ||||||||||||
\(\ds \qquad \ \ \) | \(\ds \) | \(=\) | \(\ds 1 + 10 + 36 + 56 + 35 + 6\) |
Second Edition
- Lucas discovered a relationship between Fibonacci numbers and the binomial coefficients:
- $F_{n + 1} = \dbinom n 0 + \dbinom {n - 1} 1 + \dbinom {n - 2} 1 + \cdots$