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Difference between revisions of "Fibonacci numbers"
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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 [[Composition (combinatorics)|compositions]]); there are <math>|''F''<sub>''n''+1</sub></math> ways to do this. 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: == | ||