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A349576 Recurrence a(1) = 1, a(2) = 5; a(n) = (a(n-1) + a(n-2))/GCD(a(n-1),a(n-2)) + 1.

%I A349576 #88 Aug 12 2022 19:03:20
%S A349576 1,5,7,13,21,35,9,45,7,53,61,115,177,293,471,765,413,1179,1593,309,
%T A349576 635,945,317,1263,1581,949,2531,3481,6013,9495,15509,25005,40515,4369,
%U A349576 44885,49255,18829,68085,86915,31001,117917,148919,266837,415757,682595,1098353
%N A349576 Recurrence a(1) = 1, a(2) = 5; a(n) = (a(n-1) + a(n-2))/GCD(a(n-1),a(n-2)) + 1.
%C A349576 Conjecture: this sequence is not periodic. (Note that the analogous sequences with a(2) = 1, 2, 3, or 4 are periodic.)
%H A349576 Mathematics Stack Exchange user Augusto Santi, <a href="https://math.stackexchange.com/q/4344652/121988">Investigating the recurrence relation</a>.
%o A349576 (Python)
%o A349576 from itertools import islice
%o A349576 from math import gcd
%o A349576 def A349576_gen(): # generator of terms
%o A349576     blist = [1, 5]
%o A349576     yield from blist
%o A349576     while True:
%o A349576         blist = [blist[1],sum(blist)//gcd(*blist) + 1]
%o A349576         yield blist[-1]
%o A349576 A349576_list = list(islice(A349576_gen(),30)) # _Chai Wah Wu_, Jan 10 2022
%K A349576 nonn
%O A349576 1,2
%A A349576 _Peter Kagey_, Dec 30 2021