Example Predecessor Tree for (3*n+3j)/2i (with j=12)

This is the predecessor tree (presented as a general tree) for (3*n+3j)/2i for j=12 built to contain the paths to all the odd integers from 1 to 201. The complete paths are traced only to the extent that branches determining each path have been negotiated. At that point a parenthesized number indicates the number of steps to the odd integer in question. Only the path to 21 requires branching at every level until it is reached. When the odd number in question is 0[3], the paths continuing to leaf nodes are not shown; such path continuations (with smallest i) are listed at the end of the presentation of the tree.

The levels in the tree are column aligned up to the point where indication of branching was abandoned . The root is near the bottom at the extreme left. A forward slash (/) indicates that the parent appears in a line below its appearance and a back slash (\) indicates that the parent appears in a line above its appearance. Where the branching is dense the tree is easy to follow, but where children are well separated from their parents careful attention to the column alignment is required to make the proper associations. Note the number of children (high degree of branching) sometimes utilized in this small sample, e.g. those of 59049, with 5, right near the top.

                                /  102789 --(7)-->91
                         /    6561
                                \  942597 --(5)-->63
                  /  137781
                         /  124659 --(8)-->185
                  /  452709
                         \  426465 --(7)-->159
           /   59049
                                /  185895 --(4)-->189
                         /  544563
                                                                          /     21  yes!
                                                                   /   33219
                                                            /  315549
                                                                   \  664317 --(2)-->7
                                                     /  184761
                                              /  271431
                                       /  672867
                                \ 1275021
                  \ 1082565
                  \ 2342277 --(8)-->201    (what about 73??)
                                /  172773 --(7)-->187
                         /   32805
                                              /  248103 --(4)-->177
                                              /  673353 --(5)-->131
                                       /   637875
                                              \ 1523853 --(5)-->59
                                \ 1222533
                         /  242757 --(8)-->13
                  /   19683
                                /  264627 --(7)-->53
                         \  662661
                                \  706401 --(6)-->87
                         \ 1502469 --(7)-->141
                                              /   14337 --(4)-->3
                                       /  143613
                                              \  588789 --(5)-->1
                                /  120285
                         /  111537
                                \  417717w --(7)-->83
                         /  400221 --(8)-->127
                  /  216513
                         \  977589 --(7)-->75
           /  295245
                         /  229635 --(7)-->183
                                                            /  182709 --(2)-->93
                                                     /   67473
                                                            \ 1262277 --(3)-->31
                                              /  183465
                                       /  270459
                                /  671409
                                       \  718065 --(6)-->137
                         /  636417
                  \  610173
                         \ 3077109 --(8)-->95
    /  177147
                         /   45927 --(8)-->29
                                       /   64881 --(4)-->99
                                /  181521
                                       \  306909 --(6)-->193
                         /  269001
                                \  540189 --(7)-->5
                  /  334611
                         \  715149 --(7)-->195
           \  767637
                                /  535815 --(7)-->169
                         / 1069443
                                \ 1248777 --(7)-->125
                  \ 1869885
                                /  133407 --(7)-->173
                         /  465831
                                                            /   58131 --(2)-->33
                                                     /  352917
                                                            /  763965 --(3)-->11
                                              /   99387
                                       /  414801
                                \  443961
                  /  964467
                         \ 1108809 --(8)-->79
           \ 1712421
                         / 2630961 --(7)-->105
                  \ 2106081
                                       /  250047 --(6)-->145
                                /  640791
                                       \ 1531629 --(6)-->133
                         \ 1226907
                                                            /   33237 --(1)-->117
                                                     /   39447
                                                            \  664389 --(2)-->39
                                              /  324891
                                       /  753057
                                /  697653
                         /  164025
                                              /   13365 --(4)-->57
                                       /   35721
                                              \  584901 --(5)-->19
                                /  159651
                         /  505197
                  /  255879
                                /  614547 --(3)-->81
                         \ 1187541
                                \ 1406241 --(6)-->69
           /  649539
                                /   10935 --(7)-->67
                         /  282123
                                \ 2832165 --(5)-->153
                         /  741393 --(8)-->167
                  \  688905
                         \ 1659933 --(8)-->191
                                /  422091 --(6)-->147
                         /  8988572
                                \ 1021329 --(7)-->115
                  /  807003
                         \  360855 --(8)-->163
           / 1476225
                  \ 1791153 --(7)-->9
    / 1240029
           \ 3129597  --(10)-->97
                                       /   85293 --(8)-->179
                                /   98415
                                       \  872613 --(8)-->77
                                                                          /   20763 --(2)-->51
                                                                   /  296865
                                                                          \  614493 --(3)-->17
                                                            /  355509
                                                     /   99873
                                              /  207765
                                       /   72171
                                       /  321489 --(8)-->41
                                /  373977
                                       \  820125 --(5)-->135
                         /  413343
                                              /  408969 --(4)-->27
                                       /  439587
                                              \  995085 --(7)-->113
                                \  925101
                                \ 2027349 --(9)-->107
                                / 2735937 --(9)-->109
                         / 2184813
                                \ 5649021 --(9)-->175
                  /  885735
                                       /  478953 --(7)-->111
                                /  492075
                                       \ 1135053 --(8)-->121
                         \ 1003833
                                              /   94041 --(7)-->55
                                              /  365229 --(7)-->139
                                       /  203391
                                              \  907605 --(6)-->165
                                /  570807
                                       \ 1345005 --(8)-->197
                                / 1318761 --(9)-->143
                         / 1121931
                                \ 5806485 --(9)-->151
                  / 1948617
                                / 3050865 --(9)-->37
                         \ 2421009
                                \ 6278877 --(7)-->45
           / 1594323
                  \ 4074381 --(11)-->61
                  \ 8325909
                                / 3405159 --(9)-->47
                         \ 5373459
                                \ 6987465 --(9)-->119
                                /  623295 --(8)-->149
                         / 1200663
                                \ 1423737 --(8)-->157
                  / 2066715
                         \ 2578473 --(9)-->199
           / 3365793
    / 2657205
                  /  428675 --(9)-->171
           \ 6908733
                                                                          /  801171 --(6)-->161
                                                                   / 1467477
                                                                          \ 1779489 --(6)-->101
                                                            /  308367
                                                            /  793881 --(8)-->155
                                                     /  728271
                                                                                        /  220077 --(3)-->129
                                                                                  / 148959
                                                                                        \ 1411749 --(4)-->43
                                                                          /  489159
                                                                   /  999459
                                                            \ 1764909
                                                            \ 3706965 --(8)-->35
                                              / 1358127
                                       /  2302911
                                              \ 2893401 --(9)-->123
                                              / 3011499 --(11)-->181
                                              / 6200145 --(11)-->23
                                       /  4782969
                                              \12577437 --(11)-->89
                                / 3720087
                                       \  9743085 --(11)-->25
                                                                          / 9861183 --(10)-->49
                                                                   /15057495
                                                                          \19899513 --(9)-->15
                                                            /22851963
                                                     /34543665
                                              /26040609
                                       \ 19663317
                         / 5845851
                                \15411789 --(14)-->85
                  \ 9034497
531441
    \ 5491557
                         / 1253151 --(8)-->71
                  / 2145447
                         \ 2683449 --(8)-->103

           \ 3483891
    \11160261
           \ 7263027 --(11)-->65

Since the odd integers from 1 to 201 are not necessarily leaf nodes, but the above predecessor tree stops at leaf nodes, I append a list of the remaining steps to leaf nodes for the sake of completeness. For 0[9], 2 or more steps are required; for 3[9] and 6[9], a single step suffices.

from    steps to    from    steps to                         from    steps to
3       84997       9       19461, 30437                     15      150533
21      281605      27      117765, 136893, 187901           33      3077
39      35845       45      68613, 5821                      51      101381
57      134149      63      166917, 45409                    69      11269
75      27653       81      265221, 176481, 58161, 133045    87      60421
93      76805       99      93189, 71357                     105     109573
111     125957      117     142341, 12641                    123     158725
129     175109      135     7173, 128901, 166589             141     23557
147     23557       153     31749, 161509                    159     39941
165     48133       171     56325, 123253                    177     64517
183     72709       189     80901, 38589, 28661              195     89093
201     97285


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