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Fibonacci numbers: Difference between revisions
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:<math>0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots</math> | :<math>0,\;1,\;1,\;2,\;3,\;5,\;8,\;13,\;21,\;34,\;55,\;89,\;144,\; \ldots</math> | ||
Fibonacci numbers are named after Italian mathematician Leonardo of Pisa, later known as Fibonacci. In his 1202 book Liber Abaci, Fibonacci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. | |||
==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. | ||
Fibonacci numbers | 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> | |||
== Resources: == | == Resources: == | ||