This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A382590 #20 Jun 09 2025 09:11:54 %S A382590 1,2,3,5,8,18,20,896,27072,32814080,-149545811968, %T A382590 160091119521808515072,738655358988798463192241725767680, %U A382590 12485440430502138868848264866306550045930296006672384,-147240441301035233185124372803468937068922727279777614523229030890174062704096331169792 %N 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. %C A382590 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. %H A382590 Bryle Morga, <a href="https://mathoverflow.net/questions/490330/peculiar-family-of-recurrence-formula-where-for-n1-if-you-take-the-n-th-pri">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</a>, MathOverflow. %o A382590 (Python) %o A382590 from itertools import islice %o A382590 def agen(): # generator of terms %o A382590 a, b = [1, 2], [1, 1] %o A382590 while True: %o A382590 yield a[-2] %o A382590 a, b = [a[-1], a[-1]*b[-2]+a[-2]*b[-1]], [b[-1], a[-1]*b[-2]-a[-2]*b[-1]] %o A382590 print(list(islice(agen(), 15))) # _Michael S. Branicky_, Jun 02 2025 %K A382590 sign %O A382590 0,2 %A A382590 _Bryle Morga_, Mar 31 2025