strict partitions into catalans and powers of 3

[wouter meeussen]

Any number can be so partitioned in at least one way. (true?)
the following numbers can be so partitioned in one way only:
1,2,4,7,13,21,40,63,121,195,364,624,1093,1353,1794,2054,3280,3540,3981,4241,
4710,5486,5955,6215

1
2
1+3
2+5
1+3+9
2+5+14
1+3+9+27
2+5+14+42
1+3+9+27+81
...
but watch out for the strong law of small numbers: it doesn't continue like
that; after a while, the cat's and powers of 3 mix:
r[1],
q[2],
r[1] r[3],
q[2] q[5],
r[1] r[3] r[9],
q[2] q[5] q[14],
r[1] r[3] r[9] r[27],
q[2] q[5] q[14] q[42],
r[1] r[3] r[9] r[27] r[81],
q[2] q[5] q[14] q[42] q[132],
r[1] r[3] r[9] r[27] r[81] r[243],
q[2] q[5] q[14] q[42] q[132] q[429],
r[1] r[3] r[9] r[27] r[81] r[243] r[729],
q[2] q[5] q[14] q[42] q[132] q[429] r[729],
r[1] q[1430] r[3] r[9] r[27] r[81] r[243],
q[2] q[5] q[14] q[42] q[132] q[429] q[1430],
r[1] r[3] r[9] r[27] r[81] r[243] r[729] r[2187],
q[2] q[5] q[14] q[42] q[132] q[429] r[729] r[2187],
r[1] q[1430] r[3] r[9] r[27] r[81] r[243] r[2187],
q[2] q[5] q[14] q[42] q[132] q[429] q[1430] r[2187],
r[1] q[1430] r[3] r[9] r[27] r[81] r[243] r[729] r[2187],
q[2] q[5] q[14] q[42] q[132] q[429] q[4862],
r[1] q[4862] r[3] r[9] r[27] r[81] r[243] r[729],
q[2] q[5] q[14] q[42] q[132] q[429] q[4862] r[729]

Not yet OEIS-ready, and not so evident, I think.
Maybe more in the leage of Christian Bower or Vladeta Jovovic ?

Wouter.