Let $x_{n} \in \mathbb Z$ be the $n$-th term of the recurrence relation

$$ x_{n+1} = \frac{x_{n} + x_{n-1}}{(x_{n},x_{n-1})} + 1$$

where $(x_{n},x_{n-1})$ is the gcd of $x_{n}$ and $x_{n-1}$.

Some examples:

$1, 1, 3, 5, 9, 15, 9, 9, 3, 5, 9, 15,\dots \qquad $ (periodic sequence)

$1, 12, 14, 14, 3, 18, 8, 14, 12, 14, 14, 3, \dots \qquad$ (periodic sequence)

$3, 1, 5, 7, 13, 21, 35, 9, 45, 7, 53, 61, 115, 177, 293, 471, 765, 413, \dots \qquad$ (not periodic?)

If $x_{0}$ is even or $x_{1}$ is even, the sequence seems to be always periodic: how to prove it?

More difficult, I suppose, is to predict the character of the sequence (periodic or not periodic), given its initial values $x_{0}$ and $x_{1}$.

**Addendum**

The recurrence relation

$$ x_{n+1} = \frac{x_{n} + x_{n-1}}{(x_{n},x_{n-1})} + 3$$

admits the following periodic sequence (period equal to 81):

$2, 5, 10, 6, 11, 20, 34, 30, 35, 16, 54, 38, 49, 90, 142, 119, 264, 386, 328, 360, 89, 452, 544, 252, 202, 230, 219, 452, 674, 566, 623, 1192, 1818, 1508, 1666, 1590, 1631, 3224, 4858, 4044, 4454, 4252, 4356, 2155, 6514, 8672, 7596, 4070, 5836, 4956, 2701, 7660, 10364, 4509, 14876, 19388, 8569, 27960, 36532, 16126, 26332, 21232, 11894, 16566, 14233, 30802, 45038, 37923, 82964, 120890, 14564, 6160, 474, 3320, 1900, 264, 544, 104, 84, 50, 70, 15, 20, 10, 6, 11, 20, \dots$

The recurrence relation

$$ x_{n+1} = \frac{x_{n} + x_{n-1}}{(x_{n},x_{n-1})} + 17$$

with $x_0=1$ and $x_1=4$ produces the following periodic sequence (period equal to 422):

The recurrence relation

$$ x_{n+1} = \frac{x_{n} + x_{n-1}}{(x_{n},x_{n-1})} + 199$$

with $x_0=1$ and $x_1=2$ produces the following periodic sequence (period equal to 2920):

I made some numerical experiments. The following screenshots show some of the obtained results.

Each image consists of a 20x20 matrix $T(c)$, depending on the value of $c$, in which

- $x_0$ is the row index;
- $x_1$ is the column index;
- $t_{\,x_0,\,x_1}(c)$ represents the period of the sequence generated by the recurrence relation starting with the initial conditions $x_0,\,x_1$ (the symbol "+" stands for a not periodic sequence that diverges).

Results for $\,c=1$:

Results for $\,c=2$:

Results for $\,c=3$:

Results for $\,c=4$:

Results for $\,c=5$:

Here it seems that we have only two possible periods: one of lenght 17, the other of lenght 63. Notice that if we focus on the sequences of period equal to 17 and plot them for the following initial values $(x_0,x_1)=(2,2),\,(3,8),\,(7,12)$, we obtain the same "wave form"!

**I conjecture that this fact is always true.**

The wave form for the period of lenght 63 is completely different:

- for $(x_0,x_1)=(3,10)$ we have

- for $(x_0,x_1)=(13,6)$ we have

Results for $\,c=6$:

Results for $\,c=7$:

Results for $\,c=8$:

Results for $\,c=9$:

Results for $\,c=10$:

Results for $\,c=11$:

Results for $\,c=13$:

Results for $\,c=15$:

Results for $\,c=17$:

Results for $\,c=19$: