John and Ann go sign into Codeforces

John and his wife Ann have decided to go to codeforces. On the first day Ann will do one problem and John - he wants to know how it is working - 0 problem.

Let us call a(n) - and j(n) - the number of problems done by Ann - and John - at day n. We have a(0) = 1 and in the same manner j(0) = 0.

They have chosen the following rules:

On day n the number of problems done by Ann should be n minus the number of problems done by John at day t, t being equal to the number of problems done by Ann herself at day n - 1

On day n the number of problems done by John should be n minus the number of problems done by Ann at day t, t being equal to the number of problems done by John himself at day n - 1

Whoops! I think they need to lay out a little clearer exactly what there are getting themselves into!

Could you write:

functions ann(n) and john(n) that return the list of the number of problems Ann/John does on the first n days;
functions sum_ann(n) and sum_john(n) that return the total number of problems done by Ann/John on the first n days

---------example-----------
john(11)  -->  [0, 0, 1, 2, 2, 3, 4, 4, 5, 6, 6]
ann(6)    -->  [1, 1, 2, 2, 3, 3]

sum_john(75)  -->  1720
sum_ann(150)  -->  6930
=====================================================================================================================================****//optimazation should be needed//**

#solution:
 #main formula n-a(t) where t is j(n-1)
from functools import lru_cache
import time
@lru_cache ( maxsize = None)
def jon(n):
    if n==0:
        return 0
    else:
        return (n-anna(jon(n-1)))


    #main formula n-a(t) where t is j(n-1)
@lru_cache ( maxsize = None)
def anna(n):
    if n==0:
        return 1
    else:
        return (n-jon(anna(n-1)))

def john(n):
    return [jon(x) for x in range(n)]
def ann(n):
    return [anna(x) for x in range(n)]
def sum_john(n):
    return(sum(john(n)))
def sum_ann(n):
    return(sum(ann(n)))

# print(john(num))
# print(sum_john(num))
# print(ann(num))
begin=time.time()
sum_ann(32322)
end = time.time()
print(end-begin)