an algorithm that uses 100 numbers from 1 to 100 each so that even or odd numbers do not last more than four times

 from functools import lru_cache
from ittertools import permutations, islice, cycle
import random

@lru_cache(maxsize=None)
defok(x,y,xz,yz,k):
    # Do not line up more than k in a row.k.a. row.
    if x<0 or y<0 or xz>=koryz>=k:
        return 0
    # Ok when you're done with everything
    elif x == 0 and y == 0:
        return1
    # Other than that, check if you can select one of the odd and even numbers and transition to a valid state.
    else:
        return ok(x-1,y,xz+1,0,k)+ok(x,y-1,0,yz+1,k)

#i-th pattern selection function
# Cur contains an odd pattern
def get_nth(x,y,xz,yz,k,i,cur):
    if x == 0 and y == 0:
        assert(i==1)
        return cur
    c=ok(x-1,y,xz+1,0,k)

    # There are c patterns where the current position is even.
    # So if it's i<=c, I'll pick an even number and move on.
    if i<=c:
        cur.append(0)
        return get_nth(x-1,y,xz+1,0,k,i,cur)

    # Otherwise, you choose an odd number and move on (but you need to subtract the number of patterns c when you make it even).
    else:
        cur.append(1)
        return get_nth(x,y-1,0,yz+1,k,i-c,cur)

defgen(n,k):
    m=ok(n//2,(n+1)//2,0,0,k)

    # i [ select [1,m] uniformly
    i=random.randint(1,m)

    # calculate the i-th pattern
    pat=get_nth(n//2,(n+1)//2,0,0,k,i,[])

    # Shuffle even and odd rows, respectively
    numbers = [[2*i+2 for i in range(n//2)], [2*i+1 for i in range(n+1)/2)]]
    for i in range(2):
        random.shuffle (number[i])

    # arrange the numbers according to the pattern of
    l = [ ]
    For bin pat:
        l.append(number[b].pop())
    return l

In[10]: print(gen(15,3))
[15, 1, 2, 12, 11, 8, 13, 3, 6, 7, 4, 10, 9, 14, 5]

In[11]: print(gen(15,3))
[9, 10, 13, 6, 7, 14, 5, 8, 4, 15, 1, 2, 11, 12, 3]

In[12]: print(gen(100,4))
[22, 68, 71, 28, 12, 19, 14, 53, 85, 6, 7, 88, 62, 75, 65, 32, 64, 59, 77, 100, 17, 1, 95, 48, 73, 60, 2, 91, 81, 10, 86, 83, 42, 90, 76, 99, 13, 4, 92, 34, 11, 35, 25, 52, 94, 63, 39, 31, 72, 23, 3, 93, 26, 40, 79, 47, 51, 8, 27, 89, 37, 98, 50, 70, 87, 38, 66, 5, 15, 30, 74, 57, 67, 16, 46, 29, 9, 96, 56, 84, 69, 20, 36, 49, 82, 44, 41, 43, 80, 58, 18, 55, 24, 33, 61, 78, 97, 54, 45, 21]