Re: Course-of-value recursion by defining a sequence as a self-referential infinite list

If I may, I'd like to reference the paper by Jerzy Kaczmarczuk: "Generating
power of lazy semantics":
https://karczmarczuk.users.greyc.fr/arpap/lazysem.pdf

It defines much more than Fibonacci numbers using similar techniques and
beyond.

Techniques there allow expression of multibody dynamics from Lagrangian.
This gives a solution to such problems with time power series. This is not
a closed form solution, yet very useful one.

вс, 19 янв. 2025 г. в 15:45, Vanessa McHale <[email protected]>:

> Laziness turns out to allow course-of-value recursion where one might use
> memoization in other languages. But I hadn’t seen this articulated!
>
> Famously, one can use this to define the Fibonacci numbers, viz.
>
> fibs :: [Integer]
> fibs = 1 : 1: zipWith (+) fibs (tail fibs)
>
> Or the Catalan numbers:
>
> catalan :: [Integer]
> catalan = 1 : 1 : [ sum [ (-1)^(k+1) * (pc (n - ((k*(3*k-1)) /. 2)) + pc
> (n - ((k*(3*k+1))/.2))) | k <- [1..n] ] | n <- [2..] ]
>   where
>     pc m | m >= 0 = part !! m | otherwise = 0
>
>     infixl 6 /.
>     (/.) = quot
>
> I wrote up the example: http://vmchale.com/static/serve/Comb.pdf
>
> Reinhard Zumkeller has a lot of examples on OEIS: https://oeis.org/A000081
>
> Cheers,
> Vanessa McHale
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