Difference between revisions of "Binet's formula"
From specialfunctionswiki
(Created page with "==Theorem== The following formula holds: $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ where $F_n$ denotes a Fibonacci number and $\phi$ denotes th...") |
|||
| (2 intermediate revisions by the same user not shown) | |||
| Line 2: | Line 2: | ||
The following formula holds: | The following formula holds: | ||
$$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ | $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ | ||
| − | where $F_n$ denotes a [[Fibonacci | + | where $F_n$ denotes a [[Fibonacci numbers|Fibonacci number]] and $\phi$ denotes the [[golden ratio]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| + | * {{PaperReference|On a General Fibonacci Identity|1965|John H. Halton|prev=Fibonacci numbers|next=F(-n)=(-1)^(n+1)F(n)}} | ||
| + | * {{PaperReference|The Fibonacci Zeta Function|1976|Maruti Ram Murty|prev=Fibonacci zeta function|next=Fibonacci zeta in terms of a sum of binomial coefficients}} | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 22:32, 25 May 2017
Theorem
The following formula holds: $$F_n = \dfrac{\phi^n - (-\phi)^{-n}}{\sqrt{5}},$$ where $F_n$ denotes a Fibonacci number and $\phi$ denotes the golden ratio.
