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Fibonacci numbers: Difference between revisions
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==Mathematics== | ==Mathematics== | ||
The ratio \( \frac {F_n}{F_{n+1}} \ \) approaches the | The ratio \( \frac {F_n}{F_{n+1}} \ \) approaches the golden ratio as \(n\) approaches infinity. | ||
[[File:PascalTriangleFibanacci.png|thumb|right|360px|The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of [[Pascal's triangle]].]] | [[File:PascalTriangleFibanacci.png|thumb|right|360px|The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of [[Pascal's triangle]].]] | ||
The Fibonacci numbers occur in the sums of "shallow" diagonals in | The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle. | ||
:$$F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}$$ | :$$F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}$$ | ||
Counting the number of ways of writing a given number \(n\) as an ordered sum of 1s and 2s (called | Counting the number of ways of writing a given number \(n\) as an ordered sum of 1s and 2s (called compositions); there are \(F_{n+1}\) ways to do this. For example, if \(n = 5\), then \(F_{n+1} = F_{6} = 8\) counts the eight compositions summing to 5: | ||
$$1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2$$ | $$1+1+1+1+1 = 1+1+1+2 = 1+1+2+1 = 1+2+1+1 = 2+1+1+1 = 2+2+1 = 2+1+2 = 1+2+2$$ | ||