HP 12C Fibonacci Sequence

[Dieter]

(08-16-2017 05:16 AM)Gamo Wrote:  Here is the HP 12C program to find Fibonacci Sequence accurate up to 40

Your program implements an interesting formula that calculates the Fibonacci numbers directly instead of the usual approach with adding the two previous ones. I did not know this before, but it seems to be the Moivre-Binet formula:
\[F_n = \frac{1}{\sqrt{5}} \left [ \left ( \frac{1+\sqrt{5}}{2} \right )^n - \left ( \frac{1-\sqrt{5}}{2} \right )^n \right ] \]
A straightforward implementation does not require any registers and could be done like this:

Code:

01 ENTER
02 5
03 √x
04 1
05 +
06 2
07 /
ß8 1/x
09 x<>y
10 LstX
11 x<>y
12 y^x
13 x<>y
14 CHS
15 LstX
16 y^x
17 -
18 5
19 √x
20 /
21 INTG
22 GTO 00

(08-16-2017 05:16 AM)Gamo Wrote:  Anyone have a better solution?

Since for n≥0 the second term is always less than 1/2 it doesn't have to be calculated. Simply round \(\frac{1}{\sqrt{5}} \left (\frac{1+\sqrt{5}}{2} \right )^n\) to the nearest integer. That's what the following program does.

Code:

01 ENTER
02 5
03 √x
04 1
05 +
06 2
07 /
08 x<>y
09 y^x
10 5
11 √x
12 /
13 .
14 3
15 +
16 INTG
17 GTO 00

On the 10-digit 12C this also works correctly up to n=40. Above that the results are a bit larger than expected because the 10-digit value of \(\Phi\) is slightly high while \(\sqrt{5}\) in the denominator is slightly low. That's also why the rounding function in the above code does not add 0,5 but 0,3.

BTW, using code boxes like this instead of attaching a PDF (which is only available after login) may be the better solution.

Dieter