OFFSET
1,3
COMMENTS
Suggested by A351871.
Sequence appears to diverge, but it would be nice to have a proof.
From Giorgos Kalogeropoulos, Nov 01 2022 : (Start)
Conjecture: For n >= 3775 a(n) can also be expressed in the following three ways:
1) a(n) = 1 + a(n-1) + a(n-2).
2) a(n) = 2*a(n-1) - a(n-3).
3) If A = a(3774), B = a(3772) and F = Fibonacci A000045(n),
a(n) = (A+1)*F(n-3772) - (B+1)*F(n-3774) - 1.
These three formulas only work for n >= 3775. (End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..5000 (terms 1..1002 from N. J. A. Sloane)
Michael De Vlieger, Labeled scalar plot of m = log_2(a(n)), n = 1..2^12, highlighting areas with near-zero second differences of log_2(a(n)) in red, otherwise blue. Labels are indices that begin and end a run of second differences near zero. The third run begins at n approximately 3797 but continues at least to n = 2^16. "Near-zero" means m > 10^-10.
Peter Munn, Logarithmic plot of a(n)/A005711(n). (Note that A005711 has essentially constant exponential growth.)
Mathematics Stack Exchange user Augusto Santi, A singular variant of the OEIS sequence A349576.
Giorgos Kalogeropoulos, After the term a(3773) it appears that the logarithmic graph is a straight line. This happens because the GCD of two successive terms from a(3773) and on is equal to 1. I tested all the terms up to a(10^6). If this holds to infinity then the sequence diverges. Here are the log graphs: Log plot 5000 terms, Log plot 10000 terms, Log plot 100000 terms.
MAPLE
MATHEMATICA
Nest[Append[#1, #3 + Total[#2]/#3] & @@ {#1, #2, GCD @@ #2} & @@ {#, #[[-2 ;; -1]], GCD[#[[-2 ;; -1]]]} &, {1, 1}, 48] (* Michael De Vlieger, Sep 03 2022 *)
PROG
(Python)
from math import gcd
from itertools import islice
def A355898_gen(): # generator of terms
yield from (a:=(1, 1))
while True: yield (a:=(a[1], (b:=gcd(*a))+sum(a)//b))[1]
(PARI) {a355898(N=50, A1=1, A2=1)= my(a=vector(N)); a[1]=A1; a[2]=A2; for(n=1, N, if(n>2, my(g=gcd(a[n-1], a[n-2])); a[n]=g+(a[n-1]+a[n-2])/g); print1(a[n], ", ")) } \\ Ruud H.G. van Tol, Sep 19 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Sep 01 2022
STATUS
approved