Solution counts
Vertical: digit sum (multiples of 3 omitted)
Horizontal: top left-hand digit

    |    1   2   3   4   5   6   7   8   9 | total
----+--------------------------------------+------
  2 |                                      |      
  4 |                                      |      
  5 |                                      |      
  7 |                                      |      
  8 |                                      |      
 10 |                                      |      
 11 |    3   1   1       1                 |     6
 13 |    3   6  10   1                     |    20
 14 |                                      |      
 16 |                                      |      
 17 |   51  66  71  24  16   6   4         |   238
 19 |   66  47  70  40  52  24  17   9  15 |   340
 20 |                                      |      
 22 |                                      |      
 23 |   88 137 152  77 116 118 148  61  43 |   940
 25 |   84  99  91  93 124  71  79  58  45 |   744
 26 |                                      |      
 28 |                                      |      
 29 |   22  20  37  30  66  51  32  44  34 |   336
 31 |                3  14   6   5   3   8 |    39
 32 |                                      |      
 34 |                                      |      
 35 |            1       1       2       1 |     5
 37 |                            5       3 |     8
 38 |                                      |      
 40 |                                      |      
 41 |                                      |      
 43 |                                      |      
 44 |                                      |      
----+--------------------------------------+------
tot |  317 376 433 268 390 276 292 175 149 |  2676