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Fibonacci numbers: Difference between revisions
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==Mathematics== | ==Mathematics== | ||
The ratio <math> \frac {F_n}{F_{n+1}} \ </math> approaches the golden ratio as <math>n</math> approaches infinity. | The ratio <math> \frac {F_n}{F_{n+1}} \ </math> approaches the golden ratio as <math>n</math> approaches infinity. | ||
[[File:PascalTriangleFibanacci.png|thumb|right|360px|The Fibonacci numbers are the sums of the "shallow" diagonals (shown in red) of | [[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 Pascal's triangle. | The Fibonacci numbers occur in the sums of "shallow" diagonals in Pascal's triangle. | ||
:<math>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}</math> | :<math>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}</math> | ||