I found this interesting conjecture, but maybe I'm not the first to state it. I have tested it for the first $10^4$ positive integers, but that is not a proof. Can anybody prove or disprove this conjecture?
Every positive integer can be written as the sum of 1 square number, 1 pentagonal number, and 1 hexagonal number.
Note:
Square numbers are generated by the formula, $S_{n}=n^{2}$. The first ten square numbers are:
0, 1, 4, 9, 16, 25, 36, 49, 64, 81,...
Pentagonal numbers are generated by the formula, $P_{n}=\frac{1}{2}n(3n-1)$. The first ten pentagonal numbers are:
0, 1, 5, 12, 22, 35, 51, 70, 92, 117,...
Hexagonal numbers are generated by the formula, $H_{n} = n(2n-1)$. The first ten Hexagonal numbers are:
0, 1, 6, 15, 28, 45, 66, 91, 120, 153,...
Here are the solutions for the first 10 positive integers.
Numbers = Square + Pentagon + Hexagon
1 = 0 + 1 + 0
2 = 1 + 1 + 0
3 = 1 + 1 + 1
4 = 4 + 0 + 0
5 = 0 + 5 + 0
6 = 1 + 5 + 0
7 = 1 + 5 + 1
8 = 1 + 1 + 6
9 = 4 + 5 + 0
10 = 9 + 1 + 0