Fibonacci numbers: Difference between revisions

Fibonacci numbers (edit)

554 bytes added , 19 February 2023

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 ==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.]]
+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>
-The Fibonacci numbers occur in the sums of "shallow" diagonals in [[Pascal's triangle]].
+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.  For example, if <math>n = 5</math>, then <math>F_{n+1} = F_{6} = 8</math> counts the eight compositions summing to 5:
-:<math>F_n = \sum_{k=0}^{\left\lfloor\frac{n-1}{2}\right\rfloor} \binom{n-k-1}{k}</math>
+<math>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</math>
 == Resources: ==
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 *[https://en.wikipedia.org/wiki/Fibonacci_number Fibonacci numbers]
 == Discussion: ==
+[[Category:Pages for Merging]]