To let $ a, b, c, d, e, f $ whole numbers with $ a ge c ge e> 0 $, $ b> -a $ and $ a equiv b pmod2 $, $ d> -c $ and $ c equiv d pmod $ 2, $ f> -e $ and $ e equiv f pmod2 $, If everyone $ n in mathbb N = {0,1,2, ldots } $ can be written as

$$ frac {x (ax + b)} 2+ frac {y (cy + d)} 2+ frac {z (ez + f)} 2 text {with} x, y, z in mathbb N, $$

then we say the ordered tuple $ (a, b, c, d, e, f) $ is *universally over* $ mathbb N $, For example, the triangular number set of Gauss indicates this (1,1,1,1,1,1) $ is universally over $ mathbb N $, In a recent work, I found all the candidates for tuples $ (a, b, c, d, e, f) $ universally over $ mathbb N $, When $ a b $, $ c mid d $ and $ e mid f $I have shown that many candidates are indeed universal $ mathbb N $but the following 10 candidates

begin {align} & (4,0,2,0,1,3), , (4,0,2,0,1,5), , (4,0,2,6,1,1 ), (4,0,2,6,2,0), (4,4,2,0,1,3),

(4,8,2,0,1,1), (4,8,2,0,1,3), (4,12,2,0,1,1), (6,0,2,0 , 1,3), , (6,6,2,0,1,3)

end

have not yet proved to be universal $ mathbb N $,

**Guess.** All 10 listed tuples are universally completed $ mathbb N $,

Note that (4,8,2,0,1,3) $ is universally over $ mathbb N $ if and only if an integer $ n ge3 $ can be written as $ x ^ 2 + 2y ^ 2 + z (z + 1) / 2 $ With $ x in mathbb N $ and $ y, z in mathbb Z ^ + = {1,2,3, ldots } $,

Any ideas to solve the guess?

When $ a nmid b $ or $ c nmid d $ or $ e nmid f $I showed in the same article that there are 407 candidates for such tuples $ (a, b, c, d, e, f) $ universally over $ mathbb N $, It seems that none of the 407 tuples (listed in the appendix of my work) can easily be proved to be universal $ mathbb N $,