Just to add some more combinatorical information.
Preamble: I'm used to use the following letters, which are different from yours, and I'm lazy to adapt this, please bare with me...
I use
- $N$ for the (N)umber of odd-steps (you have $k$)
- $S$ for the (S)um-of-exponents-at-$2$ (or numbers of even steps in the original Collatz-definition) (you have $t_k$)
- $A,B,C,...$ or $A_k \mid _{k=1..N}$ for the single exponents,
- with conditions $1 \le A_k \le A_\max$ and $S = \sum_{k=1}^{\small N} A_k$
- where $A_\max = S- N+1$
I looked empirically for the number of possible sets of exponents $A_k$ with the restriction that $S=\lceil N \log_2(3) \rceil$ - that means the sets of orbit-candidates which must been tested for cyclicity.
Here, rotations of the exponents, for example $A,B,C,D$ and $B,C,D,A$, are taken as duplicate list entries of a cycle-candidate and are only inserted as one instance in the final list.
Here is the empirical list of sets of cycle-candidates for $N=2 .. 8$:
N S minA maxA c sets of exponents A_k for orbits to be tested
----------------------------------------------------------------------------------
[ 2, 4, 1, 3] --- 2 [1, 3][ 2, 2]
[ 3, 5, 1, 3] --- 2 [1, 1, 3][ 1, 2, 2]
[ 4, 7, 1, 4] --- 5 [1, 1, 1, 4][ 1, 1, 2, 3][ 1, 1, 3, 2][ 1, 2, 1, 3][1, 2, 2, 2]
[ 5, 8, 1, 4] --- 7 [1, 1, 1, 1, 4][ 1, 1, 1, 2, 3][ 1, 1, 1, 3, 2][ 1, 1, 2, 1, 3][ 1, 1, 2, 2, 2][ 1, 1, 3, 1, 2][ 1, 2, 1, 2, 2]
[ 6, 10, 1, 5] --- 22 ... ... ...
[ 7, 12, 1, 6] --- 66
[ 8, 13, 1, 6] --- 99
[ 9, 15, 1, 7] --- 335
[10, 16, 1, 7] --- 504
-----------------------------------------------------------
for Collatz over the negative numbers, S=ceil(N*ld3)-1 (!!)
[ 1, 1, 1, 1] --- 1 [1]
[ 2, 3, 1, 2] --- 1 [1, 2]
[ 3, 4, 1, 2] --- 1 [1, 1, 2]
[ 4, 6, 1, 3] --- 3 [1, 1, 1, 3; 1, 1, 2, 2; 1, 2, 1, 2]
[ 5, 7, 1, 3] --- 3 [1, 1, 1, 1, 3; 1, 1, 1, 2, 2; 1, 1, 2, 1, 2]
[ 6, 9, 1, 4] --- 10 [1, 1, 1, 1, 1, 4; 1, 1, 1, 1, 2, 3; ... ]
[ 7, 11, 1, 5] --- 30 [1, 1, 1, 1, 1, 1, 5; 1, 1, 1, 1, 1, ... ]
[ 8, 12, 1, 5] --- 43 [1, 1, 1, 1, 1, 1, 1, 5; 1, 1, 1, 1, ... ]
...
==================================================================
c = # orbits-to-be-tested (cyclic/repetitions removed)
My q&d-routine to detect this is extremely time-consuming; but the results and very likely the continuation of this can be found in the OEIS, hidden in the following (infinite) rectangular array (headers-lines are mine, "maxA" is reference to mine, square brackets indicate my empirical numbers $c$ of-orbits-to-be-tested):
table starts:
N:1 2 3 4 5 6 7 8 9 10 11 12 maxA
-----------------------------------------------------------------
[1,]1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... 2
1,[2, 2,] 3, 3, 4, 4, 5, 5, 6, 6, 7, ... 3
1, 2, 4, [5, 7,]10, 12, 15, 19, 22, 26, 31, ... 4
1, 3, 5, 10, 14,[22,] 30, 43, 55, 73, 91, 116, ... 5
1, 3, 7, 14, 26, 42, [66, 99,]143, 201, 273, 364, ... 6
1, 4, 10, 22, 42, 80, 132, 217,[335, 504,]728, 1038, ... 7
Update: The T()-formula in OEIS is extremely helpful!
Here is the list of systematic $c(N)$ (= for each $N$) with $N=1..29$:
$$ [1, 2, 2, 5, 7, 22, 66, 99, 335, 504, 1768, 6310, 9690, 35530, 54484, 204347, 312455, 1193010, 4552275, 7056280, 27293640, 42181080, 165056400, 644637006, 1005633632, 3964522026, 6167026726, 24512635642, 38036848410,\ldots]$$
(Here the same for Collatz-over-the-negative-numbers, $S=\lceil N \log_2(3) \rceil -1$)
$$ [1, 1, 1, 3, 3, 10, 30, 43, 143, 201, 728, 2652, 3876, 14550, 21318, 81719, 120175, 468754, 1820910, 2731365, 10752060, 16128424, 64188600, 254463276, 386782164, 1547128656, 2349343610, 9470798326, 14369476066,...]$$
$\phantom{aaaaaaaaaaaa}$ (Note that there are $3$ cycles in the negative numbers)
Your question
- "Would it follow that as k increases, the number of distinct values approaches infinity?"
is surely to be answered as "yes"; the number of candidate-orbits (even if rotations are ignored) seem to be exponentially in $N$ (your $k$).
Appendix: The Pari/GP-routines I've used is:
T(n, k) = sumdiv(gcd(n, k), d, eulerphi(d)*binomial((n+k)\d, n\d))/(n+k)
\\ this formula has been taken from OEIS-entry
l2=log(2);l3=log(3);ld3=l3/l2; \\ define common constants
{c_list=List([1]); \\ initialize a List "c_list" with first value 1
for(N=2,29 ,
S=ceil(N*ld3); \\ value for S required to allow cyclic orbit at all
maxA=S-N;
c=T(maxA,N);
listput(c_list,c); \\ append c at "c_list"
);}
print(c_list)