Fibonacci-Like Sequence With Mod

[Leroy Quet]

Let a(0) = a(1) = 1;

Let, for 2 <= m,

a(m) = a(m-1) + a(m-2) (mod m),

where 0 <= a(m) <= m-1.

So, a(m) either equals a(m-1) + a(m-2)
    or equals a(m-1) + a(m-2) - m,

depending on if a(m-1)+a(m-2) is < m or is between m and 2m-1.



The sequence begins:  1,1,0,1,1,2,3,5,0,5,5,10,3,0,3,3,6,9,15,5,0,...
(maybe)

And the sequence of m's where a(m) = 0, another interesting sequence, 
begins:

2,8,13,20,25,...
(maybe)

What can be said about this sequence? 

Does it have a closed form?

Does the second sequence (of m's where a(m) = 0) have a closed form?

Neither of the above sequences seems to be in the EIS.

thanks,
Leroy Quet