In The Edge of Deserts of Infinite Possibility
         -Origo

Cycle as: the cyclical tree-structure formed by one Aa-Bub-Hc, connecting back in on itself.
Cycle-Connection as: the Aa-Bub-Hc that form connections to multiple cycles.
Loop as: the second half-cycle formed when the first half does not close the cycle in on the same Aa-Bub-Hc as it started from.
Structure as: the set of all cycles when saturated.
         -So i called it the ¨3CLS¨

Allowing no more than one loop between the same cycles.
Not allowing loopage back to a previously visited cycle.

Cycles going from a cycle to a cycle-connection overlapp with two other structures.
Cycles going from a cycle-connection to another cycle-connection overlapp with four other structures.
And these are therefore not considered.

Structures with no loops and a cycle that have cycle-connections to more than two cycles, are likely to be duplicates of structures with one less cycle with loops.
But given interconnected chains of cycles of size three it often end up with more than one loop between the same cycles.
Structures with chains of cycles of size three and which loops, are likely to be duplicates of structures with the number of original cycles plus the number of loops minus one.
All symmetrical structures are duplicates.
         -Shackles

Allowing duplicate structures:

Let N denote the number of cycles.
Let l denote the number of loops the next cycle added to the structure can have.

For exactly l loops.
   a(N) = a(N-1) * ( (a(N-1)/a(N-2)) + (N-1)C(l) )
With all available loops.
   a(N) = a(N-1) * ( (a(N-1)/a(N-2)) + ((N-1)C(0)+(N-1)C(1)+...+(N-1)C(N-1)) )
   a(N) = a(N-1) * ( (a(N-1)/a(N-2)) + 2^(N-1) )

Let L denote the maximum number of loops the next cycle added to the structure can have.
   a(N) = a(N-1) * ( (a(N-1)/a(N-2)) + (l=0)(L)Σ((N-1)C(l)) )
   a(N) = a(N-1) * ( (a(N-1)/a(N-2)) + 2^L )

Lmax = N-1
a(1)=1, a(2)=L+1
lim(N->∞)  =>  a(N)=∞

 
Do I dare say: the Ring Theory
 

L=N-1
Filtering out duplicated structures:

   [1, 2, 7, 51, 42, ...]