First try for a Lagged, Fibonacci (pseudo) Random Number Generators

lagfib.py

#!usr/bin/python

from random import randint
from math import pow

_lag1 = 55
_lag2 = 24
_modulus = 31
#_firstterms = [randint(0,pow(2,_modulus)) for x in range(0,_lag1)]
_firstterms = [773, 744, 844, 228, 13, 1011, 691, 1, 1106, 730, 438, 102, 498, 1004, 111, 1230, 217, 1133, 703, 686, 78, 551, 60, 1009, 772, 922, 1223, 1205, 511, 876, 992, 162, 85, 296, 837, 755, 579, 268, 64, 194, 811, 645, 626, 140, 395, 1162, 322, 64, 97, 477, 117, 803, 1233, 288, 117, 594]


def lagfib(n):
    if n<=_lag1:
        # Take among the firstterms
        return _firstterms[n]
    else:
        # Compute recursively Xn
        return (lagfib(n-_lag1)+ lagfib(n-_lag2)) % int(pow(2,_modulus))

# # General form
# def lagfibG(n,lag1=55, lag2=24, modulus=31, firstterms=None):
#     print('Initialization')
#     # Chose the highest lag
#     lagmax = max(lag1, lag2)
#     if firstterms is None:
#         # Generate the first terms
#         # Warning : these first terms are generated for the initialization
#         firstterms = [randint(0, int(pow(2,modulus))) for x in range(0,lagmax)]
#         # print(firstterms)
#     print('Generating Xn')
#     if n<=lagmax:
#         return firstterms[n]
#     else:
#         # Compute recursively Xn
#         return (lagfibG(n-lag1, firstterms=firstterms) + lagfibG(n-lag2, firstterms=firstterms)) % int(pow(2,modulus))
    

Try this:

from collections import deque

def lagged_fibonacci_sequence(j: int, k: int, m: int):
    """
    A generator of a lagged Fibonacci Sequence defined by:
    ```text
    F(n) = 1, for 0 <= n < k
    F(n) = F(n - j) + F(n - k) (mod m), for n >= k.
    ```
    """
    j, k = min(j, k), max(j, k)  # allow entering offsets in any order
    d = deque([1] * k)
    while True:
        yield d[-k]
        d[-k] = (d[-j] + d[-k]) % m
        d.rotate(-1)

You can generate the first 1,000 terms of your example with:

from itertools import islice

terms = list(islice(lagged_fibonacci_sequence(55, 24, 31), 1000))

If you want to avoid the negative indices:

def lagged_fibonacci_sequence(j: int, k: int, m: int):
    """
    A generator of a lagged Fibonacci Sequence defined by:
    ```text
    F(n) = 1, for 0 <= n < k
    F(n) = F(n - j) + F(n - k) (mod m), for n >= k.
    ```
    """
    j, k = min(j, k), max(j, k)
    d = deque([1] * k)
    j -= 1
    k -= 1
    while True:
        yield d[k]
        d[k] = (d[j] + d[k]) % m
        d.rotate()