## (3n+3^{j})/2^{i}
Homologues to (3n+1)/2^{i}
Iterations

### -Collatz iteration is the j=0 case.

-j must run from 0 to infinity for the homologues.

-These homologues were not previously recognized, it seems.

-Iterations converge to 3^{j}.

-Youngest children in the j>0 case don't necessarily come at
i=1.

-Internal nodes are congruent to 0 mod 3. (0[3] are leaves in (3n+1)/2^{i})

-Leaf nodes are congruent to 1 or 2 mod 3. (These are internal in (3n+1)/2^{i})

-Tree shapes are much different, and aren't amenable to summing
densities.

-The l.d.a.s of the j=0 case **DO** appear buried within left
descents in predecessor trees for j>0 cases .

-Steps down the l.d.a.s don't necessarily reduce the power of 3 in
the d coefficient.

-Extension of odd integer m is at 2*m+3^{(j-1)}.

-Oleg
has corrected me: the homologues are (more inclusively) (3n+3*k)/2^{i} for odd k

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