Odd even Triangle Square theorem

First you have the triangular numbers:
T(i) = 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55
Then the square numbers both indexed with the first number i and j = 0
S(j) = 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100

It seems every natural number X:1...inf can be written as S(m)+T(n)+T(o) such that m+n+o is even for even X and odd for odd X... 

for example:
15 = S(3) + T(2) +T(0)
next:
16 = S(4) + T(0) +T(0)
17 = S(4) + T(1) + T(0)
18 = S(4) +T(1) +T(1)
19 = S(3) + T(4) + T(0)
there's another way to write this but so m+n+o isn't odd
20 = S(3) + T(4) + T(1)
21 = S(3) + T(3) + T(3)
22 = S(0) + T(6) + T(0)
23 = S(4) + T(3) +  T(0)
24 = S(3) + T(4) + T(3)
I don't have a proof but I think it's true for every natural number X....