Fibonacci Number as Sum of Binomial Coefficients/Mistake/First Edition

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary

$5$

Mistake

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:

\(\ds F_{12} = 144\)

\(=\)

\(\ds \paren {\frac {11} 0} + \paren {\frac {10} 1} + \paren {\frac 9 2} + \paren {\frac 8 3} + \paren {\frac 7 4} + \paren {\frac 6 5}\)

\(\ds \qquad \ \ \)

\(\ds \)

\(=\)

\(\ds 1 + 10 + 36 + 56 + 35 + 6\)

Correction

The notation is wrong.

It should read:

Lucas discovered a relationship between Fibonacci numbers and the binomial coefficients:

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

For example:

\(\ds F_{12} = 144\)

\(=\)

\(\ds \binom {11} 0 + \binom {10} 1 + \binom 9 2 + \binom 8 3 + \binom 7 4 + \binom 6 5\)

\(\ds \qquad \ \ \)

\(\ds \)

\(=\)

\(\ds 1 + 10 + 36 + 56 + 35 + 6\)

The notation was corrected for the second edition of Curious and Interesting Numbers, but a further mistake was introduced in its place.

Sources

• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $5$