All the series start with a0 = 1, 2, or 3.  We start with a0 = 1.

[1, 1 + 2k, 1 + 2k, 1 + 2k]

    [1,1,1,1] 
    [1,3,3,5]
    [1,5,5,9]
    [1,7,7,13] 
    [1,9,9,17] ...

[1, 1 + 4k, 2 + 6k, 2 + 6k]
    
    [1,1,2,2] 
    [1,5,8,8] 
    [1,9,14,14] 
    [1,13,20,20]  
    [1,17,26,26] ...

[1, 1 + 4k, 3 + 10k, 3 + 10k]

    [1,1,3,3] 
    [1,5,13,13] 
    [1,9,23,23] 
    [1,13,33,33] 
    [1,17,43,43] ...

[1, 1 + 4k, 4 + 14k, 4 + 14k]

    [1,1,4,4] 
    [1,5,18,18] 
    [1,9,32,32] 
    [1,13,46,46] 
    [1,17,60,60] ...

[1, 1 + 4k, 5 + 18k, 5 + 18k]

    [1,1,5,5] 
    [1,5,33,33] 
    [1,9,51,51] 
    [1,13,69,69] 
    [1,17,87,87] ...

[1, 2 + 2k, 2 + 2k, 3 + 4k]

    [1,2,2,3] 
    [1,4,4,7] 
    [1,6,6,11] 
    [1,8,8,15] 
    [1,10,10,19] ...

[1, 3 + 4k, 5 + 6k, 5 + 6k]

    [1,3,5,5] 
    [1,7,11,11] 
    [1,11,17,17] 
    [1,15,23,23] 
    [1,19,29,29] ...

[1, 3 + 4k, 8 + 10k, 8 + 10k]

    [1,3,8,8] 
    [1,7,18,18] 
    [1,11,28,28] 
    [1,15,38,38] 
    [1,19,48,48] ...

[1, 3 + 4k, 11 + 14k, 11 + 14k]

    [1,3,11,11] 
    [1,7,25,25] 
    [1,11,39,39] 
    [1,15,53,53] 
    [1,19,67,67] ...

[1, 3 + 4k, 14 + 18k, 14 + 18k]

    [1,3,14,14] 
    [1,7,32,32] 
    [1,11,50,50]
    [1,15,68,68] 
    [1,19,86,86] ...

[ 2, 1 + 4k, 1 + 4k, 1 + 8k]

    [2,1,1,1]
    [2,5,5,9]
    [2,9,9,17]
    [2,13,13,25]
    [2,17,17,33] ...

[ 2, 1 + 6k, 1 + 6k, 2 + 6k]

    [2,1,1,2]
    [2,7,7,8]
    [2,13,13,14]
    [2,19,19,20]
    [2,25,25,26] ...

[ 2, 1 + 6k, 1 + 6k, 1 + 12k]

    [2,1,1,1]
    [2,7,7,13]
    [2,13,13,25]
    [2,19,19,37]
    [2,25,25,49] ...

[ 2, 1 + 8k, 1 + 8k, 1 + 16k]

    [2,1,1,1]
    [2,9,9,17]
    [2,17,17,33]
    [2,25,25,49]
    [2,33,33,65] ...

[ 2, 1 + 8k, 1 + 8k, 3 + 16k]

    [2,1,1,3]
    [2,9,9,19]
    [2,17,17,35]
    [2,25,25,51]
    [2,33,33,67]

[2, 3 + 6k, 3 + 6k, 4 + 6k]

    [2,3,3,4]
    [2,9,9,10]
    [2,15,15,16]
    [2,21,21,22]
    [2,27,27,28]

[2, 3 + 4k, 3 + 4k, 5 + 8k]

    [2,3,3,5]
    [2,7,7,13]
    [2,11,11,21]
    [2,15,15,29]
    [2,19,19,37]

[2, 3 + 6k, 3 + 6k, 5 + 12k]

    [2,3,3,5]
    [2,9,9,17]
    [2,15,15,29]
    [2,21,21,41]
    [2,27,27,53]

[2, 3 + 8k, 3 + 8k, 5 +  16k]

    [2,3,3,5]
    [2,11,11,21]
    [2,19,19,37]
    [2,27,27,53]
    [2,35,35,69]

[2, 3 + 8k, 3 + 8k, 7 +  16k]
 
    [2,3,3,7]
    [2,11,11,23]
    [2,19,19,39]
    [2,27,27,55]
    [2,35,35,71]

[2, 5 + 6k, 5 + 6k, 6 +  6k]

    [2,5,5,6]
    [2,11,11,12]
    [2,17,17,18]
    [2,23,23,24]
    [2,29,29,30]

[2, 5 + 6k, 5 + 6k, 9 + 12k]

    [2,5,5,9]
    [2,11,11,21]
    [2,17,17,33]
    [2,23,23,45]
    [2,29,29,57]

[2, 5 + 8k, 5 + 8k, 11 + 16k]

    [2,5,5,11]
    [2,13,13,27]
    [2,21,21,53]
    [2,29,29,69]
    [2,37,37,85]

[2, 5 + 8k, 5 + 8k, 9 + 16k]

    [2,5,5,9]
    [2,13,13,25]
    [2,21,21,41]
    [2,29,29,57]
    [2,37,37,73]

[2, 7 + 8k, 7 + 8k, 13 + 16k]

    [2,7,7,13]
    [2,15,15,29]
    [2,23,23,45]
    [2,31,31,61]
    [2,39,39,77]

[2, 7 + 8k, 7 + 8k, 15 + 16k]

    [2,7,7,15]
    [2,15,15,31]
    [2,23,23,47]
    [2,31,31,63]
    [2,39,39,79]

[3, 1 + 6k, 2 + 6k, 2 + 12k]

    [3,1,2,2]
    [3,7,8,14]
    [3,13,14,26]
    [3,19,20,38]
    [3,25,26,50]

[3, 1 + 9k, 1 + 9k, 2 + 9k]

    [3,1,1,2]
    [3,10,10,11]
    [3,19,19,20]
    [3,28,28,29]
    [3,37,37,38]

[3, 2 + 6k, 5 + 12k, 6 + 18k]
     
     [3,2,5,6]
     [3,8,17,24]
     [3,14,29,42]
     [3,20,41,60]
     [3,26,53,78]

[ 3, 5 + 6k, 11 + 12k, 15 + 18k]

     [3,5,11,15]
     [3,11,23,33]
     [3,17,35,51]
     [3,23,47,69]
     [3,29,59,87]

[3, 2 + 3k, 2 + 6k, 3 + 9k]

     [3,2,2,3]
     [3,5,8,12]
     [3,8,14,21]
     [3,11,20,30]
     [3,14,26,39]

[3, 2 + 9k, 2 + 9k, 3 + 9k]

     [3,2,2,3]
     [3,11,11,12]
     [3,20,20,21]
     [3,29,29,30]
     [3,38,38,39]


[3, 2 + 6k, 3 + 6k, 4 + 12k]

     [3,2,3,4]
     [3,8,9,16]
     [3,14,15,28]
     [3,20,21,40]
     [3,26,27,52]

[3, 4 + 6k, 5 + 6k, 8 + 12k]
     
    [3,4,5,8]
    [3,10,11,20]
    [3,16,17,32]
    [3,22,23,44]
    [3,28,29,56]

[3, 5 + 6k, 6 + 6k, 10 + 12k]

    [3,5,6,10]
    [3,11,12,22]
    [3,17,18,34]
    [3,23,24,46]
    [3,29,30,58]

[3, 4 + 9k, 4 + 9k, 5 + 9k]

    [3,4,4,5]
    [3,13,13,14]
    [3,22,22,23]
    [3,31,31,32]
    [3,40,40,41]

[3, 5 + 9k, 5 + 9k, 6 + 9k]

    [3,5,5,6]
    [3,14,14,15]
    [3,23,23,24]
    [3,32,32,33]
    [3,41,41,42]

[3, 7 + 9k, 7 + 9k, 8 + 9k]

    [3,7,7,8]
    [3,16,16,17]
    [3,25,25,26]
    [3,34,34,35]
    [3,43,43,44]

[3, 8 + 9k, 8 + 9k, 9 + 9k]
 
    [3,8,8,9]
    [3,17,17,18]
    [3,26,26,27]
    [3,35,35,36]
    [3,44,44,45]