I looked at the LL-sequence without the mod step:

S

_{0}=4, S

_{1}=14, S

_{2}=194, S

_{3}=37634, S

_{4}=1416317954, S

_{5}=2005956546822746114,....

For every p where M

_{p} is a Mersenne prime, S

_{p-2} is divisible by M

_{p} and by 2 (S

_{p-2} always even). I found that if you divide S

_{p-2} by 2*M

_{p} and add 1 to the remaining factor that number is divisible by 2

^{p} and the remaining factor is odd. That is:

S

_{p-2} = 2*M

_{p}*(n*2

^{p} - 1)

Where n is the remaining (big)

**odd** factor.

I only tested this for S

_{3}, S

_{5}, S

_{11}, S

_{15}, S

_{17}, S

_{29} corresponding to M5, M7, M13, M17, M19, M31, so this might be the

Law of small numbers . (Allthough S

_{29} isn't exactly "small" at 307,062,002 digits )

Next possible number to test this on is S

_{59} corresponding to M61, but at 3.3*10

^{17} digits that isn't likely.