

A248797


Number of terms > 1 in Fibonaccivariation of Collatz sequence starting with (1, 2n+1).


1



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, 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, 17, 8, 14, 1, 16, 8, 6, 5, 11, 12, 8, 8, 11, 13, 8, 9, 8, 12, 4
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OFFSET

0,3


COMMENTS

In a Fibonaccivariation 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.
Proof: Let FC be Fibonaccivariation of Collatz sequence, then FC_{n+1} <= max(FC_{n},FC_{n1}), with = only if FC_{n}=FC_{n1}. 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_{n1} 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.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..20000


EXAMPLE

a(8)=8 as the FibonacciCollatz sequence starting with 1, 17 becomes 1, 17, 9, 13, 11, 3, 7, 5, 3, 1 and has 8 terms > 1.


MAPLE

b:= proc(i, j) local m, r; m:= i+j;
while irem(m, 2, 'r')=0 do m:=r od; m
end:
a:= proc(n) local i, j, k; i, j:= 1, 2*n+1;
for k from 0 while j<>1 do i, j:= j, b(i, j) od; k
end:
seq(a(n), n=0..100); # Alois P. Heinz, Mar 15 2015


CROSSREFS

Cf. A000045, A006577, A174429.
Sequence in context: A058118 A332632 A124226 * A193536 A152290 A248699
Adjacent sequences: A248794 A248795 A248796 * A248798 A248799 A248800


KEYWORD

nonn


AUTHOR

Floor van Lamoen, Mar 03 2015


STATUS

approved



