Editing Fibonacci numbers
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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 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 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> | ||
Counting the number of ways of writing a given number <math>n</math> as an ordered sum of 1s and 2s (called compositions); there are <math>F_{n+1}</math> ways to do this.Β For example, if | Counting the number of ways of writing a given number <math>n</math> as an ordered sum of 1s and 2s (called [[Composition (combinatorics)|compositions]]); there are <math>F_{n+1}</math> ways to do this.Β For example, if {{math|1=''n'' = 5}}, then {{math|1=''F''<sub>''n''+1</sub> = ''F''<sub>6</sub> = 8}} counts the eight compositions summing to 5: | ||
:{{math|1=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}}. | |||
== Resources: == | == Resources: == | ||
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*[https://en.wikipedia.org/wiki/Fibonacci_number Fibonacci numbers] | *[https://en.wikipedia.org/wiki/Fibonacci_number Fibonacci numbers] | ||
== Discussion: == | == Discussion: == | ||