After reading the Sconibulus very interesting answer, (he is the real solver) the two sequence are
000 / 112233445566 / 01020305011 /000 (X fingers per hand)
000 / 11223344 / 010203011013 /000 (X-2 fingers per hand)
And this is my possible and alternative interpretation
Aliens are not transmitting the Fibonacci sequence, nor the PI, nor the EUler/Neper number etc. First and last part of the sequence are like CQ morse code: my call starts with CQ CQ CQ wich means "Hey I'm calling here, there is some one who is listening? Be ready for my transmission!" or "End of the message, I will not transmit anymore, now take a cofee ".
112233445566 seems to be like "Check out how many different digits we have! 6 different state!" (4 in the alternative situation). As Sconibulus pointed out, 0 is the Carrier Wave. So the signals started with to impulses (1 sec of transmission, 1 sec off, 1 sec on, 1 sec off, 1 sec on and then the values as peaks in the wave)
01020305011 becomes 1 2 3 5 11 wich in base 6 is 1 2 3 5 7 /
010203011013 becomes 1 2 3 11 13 wich in base 4 is 1 2 3 5 7.
They are the first prime numbers (except for 1) in sequence.
Conclusion: How many finger?
Assuming that humans use base 10 because have 10 fingers, assuming that clever and advanced aliens could not exist without fingers, the should have 6 (or 4) fingers. On both hands? Could be, since both are even numbers. But what about number 1 (wich is not a "real" prime number). Assuming that aliens have the same rigid behaviour of our math teachers in terms of definitions, we could read the sequence as "Look at my beautiful numerical system based on 1 hand: I am telling you that the first prime numbers are 2 3 5 7..." And, after all, who can assure us that aliens have exactly 2 hands?
One more note
My solution is wrong if you note that nor in the first neither in the second aliens use their last digit. But it could be transmitted (as a composite value) in terms of amplitude of the wave to say "I am transmitting between ZERO (C) and MAX state T (or G)" instead of "I have 1,2,3,4,5,6 fingers. 6 different states in my numerical system but I will never use the last because if I would have 10 fingers my bigger state were nine!"
Sorry for some bad english! Thank you for the question