So, we are left with these sequences' formula:

o a(n) = a(n-1) + a(n-2) if a(n-1) + a(n-2) = 2k+1 where k and n are integers

or

o a(n) = (a(n-1) + a(n-2))/2 if a(n-1) + a(n-2) = 2k where k and n are integers

Let see case by case what happens.

The third term
Hyp 1 :

> a(5) = a(4) + a(3)

<=> 5 = 7 + a(3)

<=> a(3) = -2 : But, the problem speaks a list of positive integers. So that Hyp 1 does not match.

Hyp 2 :

> a(5) = (a(4) + a(3)) / 2

<=> 10 = 7 + a(3)

<=> a(3) = 3 >>> Ok

The second term
Hyp 1 :

> a(4) = a(3) + a(2)

<=> 7 = 3 + a(2)

<=> a(2) = 4 >>> seems ok for the moment

Hyp 2 :

> a(4) = (a(3) + a(2)) / 2

<=> 14 = 3 + a(2)

<=> a(2) = 11 >>> seems ok for the moment

The first term
Here, we still have 2 possible values for a(2). We will check the validity in each case.

Hyp 1 : a(2) = 4

> a(3) = a(2) + a(1)

<=> 3 = 4 + a(1)

<=> a(1) = -1 >>> Not matching : the list must be exclusively composed of positive terms.

Hyp 2 : a(2) = 4

> a(3) = (a(2) + a(1)) / 2

<=> 6 = 4 + a(1)

<=> a(1) = 2 >>> Ok.

Hyp 3 : a(2) = 11

> a(3) = a(2) + a(1)

<=> 3 = 11 + a(1)

<=> a(1) = -8 >>> Not matching : the list must be exclusively composed of positive terms.

Hyp 4 : a(2) = 11

> a(3) = (a(2) + a(1)) / 2

<=> 6 = 11 + a(1)

<=> a(1) = -5 >>> Not matching : the list must be exclusively composed of positive terms.

Finally, we have found a(1) = 2.