Fibonacci Number as Sum of Binomial Coefficients/Mistake

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:

$F_{n + 1} = \dbinom n 0 + \dbinom {n - 1} 1 + \dbinom {n - 2} 1 + \cdots$