A problem on a type of {m,n} tree

Question

So here is the tree. For given $\left \{ m,n \right \}$. {m,n} will transform to give these elements which I will represent using a summation operator;

$$\sum^{n}_{k=1}\left \{ m-k,k \right \}$$

Take {9,3}. So In our example it becomes, $\left \{ 8,1 \right \},\left \{ 7,2 \right \},\left \{ 6,3 \right \}$. And with this we can continue on to look like this;

The Rules here are that, when either the $m$ or the $n$ values becomes $0$ or $1$, end the transformation. For example here, it ends at {8,1},{6,1},{4,1},..{0,2} etc. and there are $12$ of them.

The problem here is that, given $m,n$ where $n≤m$. Find the number of times the transformation "tree" stops.

Answer 1

Just a comment, but it won't fit in a comment box. I calculated the values for $S(m,3),\ 0\leq m<100$. Note that I did get $S(9,3)=12$ as shown in the diagram.

EDIT

I've noticed that we always have $S(m,3)=m+S(m-6),\ m\geq6$ so there will definitely be recurrence relations though there may be $6$ different ones.

0 1
1 1
2 2
3 3
4 4
5 5
6 7
7 8
8 10
9 12
10 14
11 16
12 19
13 21
14 24
15 27
16 30
17 33
18 37
19 40
20 44
21 48
22 52
23 56
24 61
25 65
26 70
27 75
28 80
29 85
30 91
31 96
32 102
33 108
34 114
35 120
36 127
37 133
38 140
39 147
40 154
41 161
42 169
43 176
44 184
45 192
46 200
47 208
48 217
49 225
50 234
51 243
52 252
53 261
54 271
55 280
56 290
57 300
58 310
59 320
60 331
61 341
62 352
63 363
64 374
65 385
66 397
67 408
68 420
69 432
70 444
71 456
72 469
73 481
74 494
75 507
76 520
77 533
78 547
79 560
80 574
81 588
82 602
83 616
84 631
85 645
86 660
87 675
88 690
89 705
90 721
91 736
92 752
93 768
94 784
95 800
96 817
97 833
98 850
99 867