%I #26 Mar 15 2015 15:14:28
%S 0,1,2,1,5,5,4,1,8,3,2,3,3,5,5,1,10,4,7,3,9,7,5,4,7,2,3,11,10,12,7,1,
%T 12,6,4,6,17,10,9,6,13,5,2,4,10,10,7,7,5,6,11,8,7,15,7,10,15,9,12,9,
%U 17,8,14,1,16,8,6,5,11,12,8,8,11,13,8,9,8,12,4
%N Number of terms > 1 in Fibonacci-variation of Collatz sequence starting with (1, 2n+1).
%C In a Fibonacci-variation of Collatz sequence the next term is the odd part of the sum of the preceding two terms. The sequence terminates when 1 is reached. All sequences with initial values {1, 2n+1} terminate.
%C Proof: Let FC be Fibonacci-variation of Collatz sequence, then FC_{n+1} <= max(FC_{n},FC_{n-1}), with = only if FC_{n}=FC_{n-1}. Therefore FC cannot get into a loop of length greater than 1 (because for all i>n FC_{i}< max(FC_{n},FC_{n+1}). When FC_{n} and FC_{n-1} are coprime, FC_{n} and FC_{n+1} are coprime as well. We conclude that with initial values 1 and 2n+1 (n>0) all pairs of consecutive terms must be coprime, hence cannot become constant (loop of length 1) different from 1.
%H Alois P. Heinz, <a href="/A248797/b248797.txt">Table of n, a(n) for n = 0..20000</a>
%e a(8)=8 as the Fibonacci-Collatz sequence starting with 1, 17 becomes 1, 17, 9, 13, 11, 3, 7, 5, 3, 1 and has 8 terms > 1.
%p b:= proc(i, j) local m, r; m:= i+j;
%p while irem(m, 2, 'r')=0 do m:=r od; m
%p end:
%p a:= proc(n) local i, j, k; i, j:= 1, 2*n+1;
%p for k from 0 while j<>1 do i, j:= j, b(i, j) od; k
%p end:
%p seq(a(n), n=0..100); # _Alois P. Heinz_, Mar 15 2015
%Y Cf. A000045, A006577, A174429.
%K nonn
%O 0,3
%A _Floor van Lamoen_, Mar 03 2015