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(10-13-2021 04:42 PM)Gerson W. Barbosa Wrote: [ -> ]RPN: Y, Z, T <- k1; X <-k2 Then * * * …
RPL: Fill the infinite stack with [ [ 1 1 ] [ 1 0 ] ] Then * * * …

and

Thank to remind me this old matrix arithmetic multiplication on RPL. I remember a few decade when I try to use power exponentiation of [[1 1][1 0]] to the n to directly get Fn value. Hard time discovering sadly that the matrix exponentiation on the HP-28C/S doesn't work with ^ instruction.

(10-13-2021 06:26 PM)Dave Britten Wrote: [ -> ]I think you guys have figured out the gist of it.
The solution in the book is to set FIX 0, and fill the stack thusly:

T: phi
Z: phi
Y: phi
X: 1/SQRT(5)

Where phi is the golden ratio: (SQRT(5)+1)/2. The 1/SQRT(5) term is derived from Binet's formula.

OK, now I can publish my completed trace print.
[attachment=9914]

(10-13-2021 07:35 PM)David Hayden Wrote: [ -> ]
(10-13-2021 07:11 PM)Gerson W. Barbosa Wrote: [ -> ]0 1 + Then repeat RCL + ST L
That's it!

I can't applied this trick neither on my HP-41C that miss memory recall arithmetic, nor on my HP-15C who have recall arithmetic but no stack register manipulation.
The amazing HP 42S (or succedanea DM 42S or Free 42) is really a blend of the two anthology systems.

At least, like the HP-25, the HP-41 and HP-15C need a two steps naked program « LASTx + », close to a simple RPL code leaving the Fibonacci sequence reversed in the stack initiate by 1 DUP and repeating « OVER OVER + » over and over...
(10-14-2021 10:03 AM)C.Ret Wrote: [ -> ]

We can use this to get fib(n+m) = fib(m)*fib(n+1) + fib(m-1)*fib(n)

If n=m, we have fib(2n) = fib(n) * (fib(n+1) + fib(n-1))
Using this, we can can 1/sqrt(5) another way

φ^n ≈ fib(2n) / fib(n) = fib(n+1) + fib(n-1) ≈ fib(n) * (φ+1/φ) = fib(n) * √5

fib(n) = round(φ^n / √5)
(10-14-2021 02:13 PM)Albert Chan Wrote: [ -> ]
(10-14-2021 10:03 AM)C.Ret Wrote: [ -> ]

We can use this to get fib(n+m) = fib(m)*fib(n+1) + fib(m-1)*fib(n)

If n=m, we have fib(2n) = fib(n) * (fib(n+1) + fib(n-1))
Using this, we can can 1/sqrt(5) another way

φ^n ≈ fib(2n) / fib(n) = fib(n+1) + fib(n-1) ≈ fib(n) * (φ+1/φ) = fib(n) * √5

fib(n) = round(φ^n / √5)

Looks like more than 1 program step to me....

Also, where are those functions on an HP-25? I've looked at mine again today, and just can't see them anywhere...
(10-13-2021 08:34 PM)rprosperi Wrote: [ -> ]Many other models support recall arithmetic (45, 27, 15C, 32S, 32SII, others?) but only the 42S includes recall arithmetic AND stack register access.

Since we’re at it, I decided to write an HP-42S program to display the Fibonacci sequence trying to use the least number of steps (Not sure whether I’ve succeeded or not – another one I tried has exactly the same number of steps and bytes). Interestingly an equivalent HP-15C program, which lacks recall stack arithmetic but does have recall arithmetic, is possible using the same number of steps (8).

Code:
``` 00 { 20-Byte Prgm } 01▸LBL "Fib" 02 2 03 -1 04 + 05▸LBL 00 06 RCL+ ST L 07 STOP 08 GTO 00 09 END```

XEQ “Fib” → 0
R/S → 1
R/S → 1
R/S → 2
R/S → 3
R/S → 5
R/S → 8
(10-16-2021 02:55 PM)Gerson W. Barbosa Wrote: [ -> ](Not sure whether I’ve succeeded or not – another one I tried has exactly the same number of steps and bytes)

Surely I hadn’t. It is possible to do it in 6 steps and 17 steps.
Thanks Mike (Stgt) for his comment
“Many think a program must start
with a global label, but no, it needs not” and examples which easily lead to

Code:
``` 00 { 17-Byte Prgm } 01 GTO 00 02▸LBL "Fib" 03 -1 04 ABS 05▸LBL 00 06 RCL+ ST L 07 END```

Edited for a minor grammar issue
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