-th iterate of 
,
as presented in our introduction,
In Theorem 1.1 of his 1975 paper [4],
Terras states that the -th iterate of ,
as presented in our introduction,
can be expressed as a "remainder representation" of the form
where 
represents 
, applied times, and 
.
He then goes to define
We shall attempt here to rewrite this "remainder representation" as the same statement
as equation 
in our introduction,
appearing in the 1978 paper [2] by Böhm and Sontacchi (as Proposition 7),
for any number 
that does arrive at 
by the iteration of the Collatz function.
Recall that we defined
an integer 
and constants 
,
with 
and 
,
that represent the steps taken by iterations of the original Collatz function 
as it goes from to :
steps of the rule 
are applied (possibly zero);
;
steps of the rule ;;
steps of the rule ;,
steps of the rule .
The constant represents the number of times the rule
was applied during the entire trajectory of from to .
We also defined constants 
as partial sums of the 
(
, 
, 
, etc.).
Back to Terras, he uses a "shortcut" and applies the mapping 
to odd numbers.
Therefore, a trajectory of 
from to is shorter (by steps)
than a similar trajectory of the original function.
(If the total steps were 
for the function ,
there are only 
steps when using the function .)
So we have 
representing the total number of applications
of the function , starting from and until reaching .
Notice that 
are precisely the 
positions
in the 
vector where it contains a value of
(the times the rule was used, plus the very last element 
corresponding to our arrival at ).
Armed with all these definitions, we have
and finally, since
we end up having
which is precisely equation above.
