Let a,b,c,d,e,f be integers with a >= c >= e > 0, b > -a, d > -c, f > -e, and a-b, c-d, e-f all even, and b >= d if a = c, and d >= f if c = e. Suppose that we don't have
(a|b & c|d & e|f). If the ordered tuple (a,b,c,d,e,f) is universal over {0,1,2,...}, i.e., every n = 0,1,2,... can be written as 

      x*(ax+b)/2 + y*(cy+d)/2 + z*(ez+f)/2     with x,y,z nonnegative integers,

then (a,b,c,d,e,f) must be in the following list of 407 ordered tuples, and we conjecture that each of the 407 listed tuples (a,b,c,d,e,f) is indeed universal over {0,1,2,...}. 
(Cf. http://arxiv.org/abs/1502.03056.)


                                                                                                                 -- Zhi-Wei Sun (July 27, 2017)




a b c d e f
-------------------------------------
3 -1 1 1 1 1
3 -1 1 3 1 1
3 -1 1 5 1 1
3 -1 1 7 1 1
3 -1 1 9 1 1
3 -1 1 11 1 1
3 -1 1 13 1 1
3 -1 1 15 1 1
3 -1 1 17 1 1
3 -1 2 0 1 1
3 -1 2 0 1 3
3 -1 2 0 1 5
3 -1 2 0 2 0
3 -1 2 2 1 1
3 -1 2 2 1 3
3 -1 2 2 2 0
3 -1 2 4 1 1
3 -1 2 4 1 3
3 -1 2 4 2 2
3 -1 2 6 1 1
3 -1 2 8 1 1
3 -1 2 10 1 1
3 -1 2 12 1 1
3 -1 2 16 1 1
3 -1 3 -1 1 1
3 -1 3 -1 1 3
3 1 1 1 1 1
3 1 1 3 1 1
3 1 1 5 1 1
3 1 1 7 1 1
3 1 2 0 1 1
3 1 2 0 1 3
3 1 2 2 1 1
3 1 2 2 2 0
3 1 2 4 1 1
3 1 2 6 1 1
3 1 2 6 2 0
3 1 3 -1 1 1
3 1 3 -1 1 3
3 1 3 -1 2 2
3 1 3 -1 2 6
3 3 3 -1 1 1
3 3 3 -1 1 3
3 3 3 -1 2 0
3 3 3 -1 2 2
3 3 3 1 1 1
3 3 3 1 2 0
3 5 1 1 1 1
3 5 1 3 1 1
3 5 2 0 1 1
3 5 2 0 1 3
3 5 2 2 1 1
3 5 2 2 2 0
3 5 3 -1 1 1
3 5 3 1 1 1
3 7 1 1 1 1
3 7 2 0 1 1
3 7 2 2 2 0
3 7 3 -1 1 1
3 9 3 -1 1 1
3 11 2 0 1 1
3 11 2 0 1 3
3 11 3 -1 1 1
3 13 2 0 1 1
3 15 3 -1 1 1
4 -2 1 1 1 1
4 -2 1 3 1 1
4 -2 1 3 1 3
4 -2 1 5 1 1
4 -2 1 5 1 3
4 -2 1 7 1 1
4 -2 1 7 1 3
4 -2 1 9 1 1
4 -2 2 0 1 1
4 -2 2 0 1 3
4 -2 2 0 1 5
4 -2 2 2 1 1
4 -2 2 2 1 3
4 -2 2 2 1 5
4 -2 2 2 2 0
4 -2 2 4 1 1
4 -2 2 8 1 1
4 -2 3 -1 1 1
4 -2 3 -1 1 3
4 -2 3 -1 1 5
4 -2 3 -1 2 0
4 -2 3 1 1 1
4 -2 3 1 1 3
4 -2 3 1 2 0
4 -2 3 7 1 1
4 0 3 -1 1 1
4 0 3 -1 1 3
4 0 3 -1 1 5
4 0 3 -1 1 7
4 0 3 -1 2 0
4 0 3 -1 2 4
4 0 3 1 1 1
4 0 3 1 2 0
4 0 3 5 1 1
4 0 3 5 2 0
4 0 4 -2 1 1
4 0 4 -2 1 3
4 0 4 -2 1 5
4 0 4 -2 3 -1
4 2 2 0 1 1
4 2 2 0 1 3
4 2 3 -1 1 1
4 2 3 -1 1 3
4 4 3 -1 1 1
4 4 3 1 1 1
4 6 3 -1 1 1
4 8 3 -1 1 1
4 10 3 -1 1 1
5 -3 1 1 1 1
5 -3 1 3 1 1
5 -3 1 5 1 1
5 -3 1 7 1 1
5 -3 1 9 1 1
5 -3 2 0 1 1
5 -3 2 0 1 3
5 -3 2 2 1 1
5 -3 2 2 1 3
5 -3 2 2 2 0
5 -3 2 4 1 1
5 -3 2 4 1 3
5 -3 2 4 2 2
5 -3 2 8 1 1
5 -3 3 -1 1 1
5 -3 3 1 1 1
5 -3 3 1 1 3
5 -3 3 3 3 -1
5 -3 3 7 1 1
5 -1 2 0 1 1
5 -1 2 0 1 3
5 -1 2 0 1 5
5 -1 2 0 1 7
5 -1 2 0 1 9
5 -1 2 2 1 1
5 -1 2 6 1 1
5 -1 3 -1 1 1
5 -1 3 1 1 1
5 -1 4 0 1 1
5 1 2 0 1 1
5 1 2 0 1 3
5 1 2 2 1 1
5 1 3 -1 1 1
5 1 3 -1 1 3
5 1 3 1 1 1
5 3 1 1 1 1
5 3 1 3 1 1
5 3 2 0 1 1
5 3 2 2 1 1
5 3 2 2 2 0
5 3 3 -1 1 1
5 3 3 1 2 0
5 3 4 -2 1 1
5 3 4 -2 1 3
5 7 2 0 1 1
5 7 2 0 1 3
5 7 3 -1 1 1
5 9 3 -1 1 1
5 11 2 0 1 1
5 13 3 -1 1 1
6 -4 1 1 1 1
6 -4 1 3 1 1
6 -4 1 3 1 3
6 -4 1 5 1 1
6 -4 1 5 1 3
6 -4 1 7 1 1
6 -4 1 7 1 3
6 -4 1 9 1 1
6 -4 2 0 1 1
6 -4 2 0 1 3
6 -4 2 2 1 1
6 -4 2 2 1 3
6 -4 2 2 2 0
6 -4 2 6 1 1
6 -4 2 6 1 3
6 -4 3 -1 1 1
6 -4 3 1 1 1
6 -4 3 1 1 3
6 -4 3 1 1 5
6 -4 3 5 1 1
6 -4 3 7 1 1
6 -4 4 0 1 1
6 -4 4 0 1 3
6 -4 4 0 1 5
6 -4 4 0 1 7
6 -4 5 -3 1 1
6 -2 1 1 1 1
6 -2 1 3 1 1
6 -2 1 5 1 1
6 -2 1 7 1 1
6 -2 2 0 1 1
6 -2 2 0 1 3
6 -2 2 0 1 5
6 -2 2 0 1 7
6 -2 2 0 1 9
6 -2 2 2 2 0
6 -2 3 1 1 1
6 -2 4 -2 1 1
6 -2 4 -2 1 3
6 -2 4 -2 1 5
6 -2 4 0 1 1
6 -2 4 4 1 1
6 -2 5 -3 1 1
6 -2 5 -3 1 3
6 0 3 -1 1 1
6 0 3 -1 2 2
6 0 3 1 1 1
6 0 6 -4 1 1
6 0 6 -4 2 2
6 2 1 1 1 1
6 2 1 3 1 1
6 2 2 2 1 1
6 2 2 2 2 0
6 2 3 -1 1 1
6 2 3 1 1 1
6 2 4 0 1 1
6 2 6 -4 1 1
6 4 1 1 1 1
6 4 2 2 2 0
6 4 6 -4 1 1
6 6 3 -1 1 1
6 8 2 0 1 1
6 8 3 -1 1 1
6 10 2 0 1 1
6 12 3 -1 1 1
7 -5 2 0 1 1
7 -5 2 2 1 1
7 -5 2 4 1 1
7 -5 2 6 1 1
7 -5 2 8 1 1
7 -5 3 -1 1 1
7 -5 3 -1 1 3
7 -5 3 -1 2 2
7 -5 3 5 1 1
7 -5 4 0 1 1
7 -3 2 0 1 1
7 -3 2 2 1 1
7 -3 2 6 1 1
7 -3 3 -1 1 1
7 -3 3 -1 1 3
7 -3 3 -1 1 5
7 -3 3 1 1 1
7 -3 4 0 1 1
7 -1 2 0 1 1
7 -1 2 0 1 3
7 -1 2 2 1 1
7 -1 3 1 1 1
7 -1 6 -4 1 1
7 1 2 0 1 1
7 1 3 -1 1 1
7 3 1 1 1 1
7 3 2 2 2 0
7 5 2 0 1 1
7 5 2 0 1 3
8 -6 1 1 1 1
8 -6 1 3 1 1
8 -6 1 5 1 1
8 -6 1 7 1 1
8 -6 2 0 1 1
8 -6 2 2 1 1
8 -6 2 2 1 3
8 -6 2 2 2 0
8 -6 2 4 2 2
8 -6 3 1 1 1
8 -6 5 -3 1 1
8 -6 5 -3 1 3
8 -4 2 2 1 1
8 -4 2 6 1 1
8 -4 6 -4 1 1
8 -2 3 -1 1 1
8 -2 3 1 1 1
8 0 3 -1 1 1
8 0 3 1 1 1
8 0 6 -2 1 1
8 2 1 1 1 1
8 2 2 2 2 0
9 -7 2 0 1 1
9 -7 2 4 1 1
9 -7 2 8 1 1
9 -7 3 -1 1 1
9 -7 5 -1 1 1
9 -7 5 -1 1 3
9 -5 2 2 1 1
9 -5 2 6 1 1
9 -5 3 -1 1 1
9 -5 3 -1 1 3
9 -5 3 5 1 1
9 -5 4 0 1 1
9 -1 1 1 1 1
9 -1 1 3 1 1
9 -1 2 0 1 1
9 -1 2 2 1 1
9 -1 2 2 2 0
9 -1 4 0 1 1
9 5 3 -1 1 1
9 7 2 0 1 1
9 9 3 -1 1 1
10 -8 1 1 1 1
10 -8 1 3 1 1
10 -8 1 5 1 1
10 -8 1 7 1 1
10 -8 1 9 1 1
10 -8 2 0 1 1
10 -8 2 2 1 1
10 -8 2 2 1 3
10 -8 2 2 2 0
10 -8 2 4 1 1
10 -8 2 8 1 1
10 -8 3 1 1 1
10 -8 3 1 1 3
10 -8 3 7 1 1
10 -6 2 0 1 1
10 -6 2 0 1 3
10 -6 2 0 1 5
10 -6 2 2 1 1
10 -6 3 -1 1 1
10 -6 3 5 1 1
10 -6 5 -3 1 1
10 -4 2 0 1 1
10 -4 2 0 1 3
10 -4 3 -1 1 1
10 -2 3 -1 1 1
10 2 3 -1 1 1
10 4 3 -1 1 1
10 6 2 0 1 1
10 8 3 -1 1 1
11 -9 2 0 1 1
11 -9 3 -1 1 1
11 -9 3 1 1 1
11 -9 3 7 1 1
11 -7 1 1 1 1
11 -7 1 3 1 1
11 -7 1 5 1 1
11 -7 2 2 2 0
11 -7 3 -1 1 1
11 -5 2 0 1 1
11 -5 2 0 1 3
11 -3 3 -1 1 1
11 -3 3 1 1 1
11 5 2 0 1 1
12 -8 2 0 1 1
12 -8 2 0 1 3
12 -8 2 2 1 1
12 -4 1 1 1 1
12 -4 1 3 1 1
12 -4 2 2 1 1
12 -4 2 2 2 0
12 -4 4 0 1 1
13 -11 2 0 1 1
13 -11 2 2 1 1
13 -11 2 2 1 3
13 -9 2 0 1 1
13 -9 2 0 1 3
13 -7 2 0 1 1
13 -7 2 0 1 3
13 -5 2 0 1 1
14 -12 2 2 1 1
14 -10 2 0 1 1
14 -10 2 0 1 3
14 -10 2 2 1 1
14 -10 3 -1 1 1
14 -2 2 0 1 1
14 -2 3 -1 1 1
14 2 2 0 1 1
15 -13 1 1 1 1
15 -13 1 3 1 1
15 -13 1 5 1 1
15 -13 1 7 1 1
15 -13 1 9 1 1
15 -13 2 0 1 1
15 -13 2 2 1 1
15 -13 2 2 1 3
15 -13 2 2 2 0
15 -13 2 4 1 1
15 -13 2 4 2 2
15 -13 2 8 1 1
15 -13 3 1 1 1
15 -13 3 7 1 1
15 -13 4 0 1 1
15 -11 1 1 1 1
15 -11 2 0 1 1
15 -11 2 2 2 0
15 -7 1 1 1 1
15 -7 2 0 1 1
15 -7 2 2 1 1
15 -7 2 2 2 0
15 -3 3 -1 1 1
15 3 3 -1 1 1
16 -14 2 0 1 1
16 -10 2 0 1 1
16 -10 2 0 1 3
16 -8 3 -1 1 1
16 -4 2 0 1 1
17 -15 3 1 1 1
17 -15 3 1 1 3
18 -10 2 0 1 1
20 -16 2 2 1 1
20 -16 2 6 1 1
20 -12 3 -1 1 1
20 -4 2 0 1 1
21 -19 2 2 1 1
21 -9 2 0 1 1
21 -5 2 0 1 1
25 -23 2 0 1 1
-----------------------------------