A382590 a(n) = a(n-1)*b(n-2) + a(n-2)*b(n-1) and b(n) = a(n-1)*b(n-2) - a(n-2)*b(n-1) starting with a(0) = b(0) = b(1) = 1 and a(1) = 2.

1, 2, 3, 5, 8, 18, 20, 896, 27072, 32814080, -149545811968, 160091119521808515072, 738655358988798463192241725767680, 12485440430502138868848264866306550045930296006672384, -147240441301035233185124372803468937068922727279777614523229030890174062704096331169792

Offset: 0

Author

Bryle Morga, Mar 31 2025

Keywords

• sign

Comments

This sequence appears to have a very peculiar (conjectured) property. For any k > 1, if you take the k-th prime factor of each term, you get an eventually periodic sequence. This seems to hold even when we change a(1) as long as it is an integer > 1. For discussion about this conjecture, there's a link to a MathOverflow question here.

Links

• Bryle Morga, Peculiar family of recurrence formula where for n>1 if you take the n-th prime factor of each term, you get an eventually periodic sequence, MathOverflow.

Programs

Python

from itertools import islice
def agen(): # generator of terms
    a, b = [1, 2], [1, 1]
    while True:
        yield a[-2]
        a, b = [a[-1], a[-1]*b[-2]+a[-2]*b[-1]], [b[-1], a[-1]*b[-2]-a[-2]*b[-1]]
print(list(islice(agen(), 15))) # Michael S. Branicky, Jun 02 2025