HP 12C Fibonacci Sequence

[C.Ret]

(08-12-2024 11:59 AM)Werner Wrote:  For a few more correct results, define the constants more accurately as
.2 (g)[√x] [CHS] [ PV ] and
200 [ENTER] 5 [√x] 1 [ + ] [ ÷ ] [ i ]
Now the first wrong result is for n=44. Mh. Doesn't gain a lot...

Thank you for this pertinent remark.
I had not noticed that my way of doing the two calculations gave results so far from the two true values. I never think about it; it is annoying with the voyagers which only have ten digits. I am unforgivable, especially for the constant having to be raised by a power.

Now, even with all the digits correctly set, it is true that the improvement is a little disappointing and that we push back the limit of this method by only four ranks.

But, well, F_44 is already not bad.
Especially since the HP-12C can no longer give all the digits beyond F_49.

Thank you for your help.

---

P.S.: By re-reading my post, I just get an idea.

Since the last exact Fibonacci Value we may obtain on an HP Voyager is F_49 = 7'778'742'049.
Why not using the ability of the HP-12C to adjust the right parameters instead of trying to randomly entering accurate or inaccurate parameters manually?

The procedure may be:
_1_ Input best approximation of both parameters (using good advice to avoid unforgivable mistaken) as initial guests.
(f)[CLEAR][ FIN ]
.2 (g)[√x] [CHS] [ PV ]
200 [ENTER] 5 [√x] 1 [ + ] [ ÷ ] [ i ]

If we test result for F_49:
49 [ n ][ FV ] display erroneous value 7'778'742'061.

_2_* Try to obtain the correct value:
49 [ n ]
7778742049 [ FV ]
Now try to alternatively optimize the two parameters PV and i% to achieve expected result.

Optimize PV by pressing the [ PV ] key. The new value for PV is displayed -0.447213595
Checking the accuracy of the new configuration:
49 [ n ] [ FV ] now display 7'778'742'047. Much better but still off by two units.

Optimize i% by reentering expected value of F_49 and pressing the [ i ] key.
49 [ n ]
7778742049 [ FV ]
[ i ] now display the new adjusted i parameter 61.80339888

Checking the accuracy of the new configuration:
49 [ n ] [ FV ] still display 7'778'742'047.
No good, just continue to adjust.

Reset system:
49 [ n ]
7778742049 [ FV ]
Optimize PV again by pressing the [ PV ] key. The same value for PV is displayed -0.447213595
But same unreachable bits may have been changed internally(*), since, if we check:
49 [ n ] [ FV ] now display correct value 7'778'742'049.

But this is not the last good news :
48 [ n ] [ FV ] display 4'807'526'976.
47 [ n ] [ FV ] display 2'971'215'073.
46 [ n ] [ FV ] display 1'836'311'903.
45 [ n ] [ FV ] display 1'134'903'170.
44 [ n ] [ FV ] display 701'408'733.
...
Good settings !

(*) holding the (f)[PREFIX] key may give you some clue here about this mystery.