How can I generate series of Pell numbers instead of a specific one in Lisp

Question

How do I use cons or other way to print a list of Pell numbers till the Nth number?

(defun pellse (k)
   (if (or (zerop k) (= k 1))
       k
   (+ (* 2 (pellse (         


        

Answer 1

Here is how to do it in a way that won’t be exponential:

(defun pells (n)
  (loop repeat n
    for current = 0 then next
    and next = 1 then (+ (* 2 next) current)
    collect current))

The time complexity to calculate the nth element given the two previous elements is O(log(P_n)) where P_n is the nth Pell number; you need log(P_n) bits for the answer and log(P_n) operations for the addition. We don’t actually need to work out what P_n is: It is defined by a simple linear recurrence relation so the solution must be exponential so log(P_n) = O(n). Therefore the complexity of calculating the first n Pell numbers is O(n*n) = O(n²).

One cannot^[*] do better than O(n²) as one must write O(n²) bits to represent all these numbers.

^[*] Although I very much doubt this, it might, in theory, be possible to represent the list in some more compact way by somehow sharing data.