HP 12C Fibonacci Sequence

[Thomas Klemm]

Here's another idea:

(08-06-2023 11:06 AM)Thomas Klemm Wrote:  Splitting the Index

But since the biggest value for \(n\) is only \(49\) we can use the following formulas to reduce the index:

\(
\begin{aligned}
F_{2n-1}&={F_{n}}^{2}+{F_{n-1}}^{2}\\
F_{2n}&=(2F_{n-1}+F_{n})F_{n}\\
\end{aligned}
\)

Now even in the worst case of \(F_{49}\) we can use rounding to calculate \(F_{24}\) and \(F_{25}\).

Examples

\(
\begin{aligned}
F_{44}&=&(2F_{21}+F_{22})F_{22} &=& (2 \cdot 10,946 + 17,711) \cdot 17,711 &=& 701,408,733 \\
F_{49}&=&{F_{25}}^{2}+{F_{24}}^{2} &=& 75,025^2 + 46,386^2 &=& 7,778,742,049 \\
\end{aligned}
\)