EACH NATURAL NUMBER IS OF THE FORM x 2 + (2y) 2 + z(z + 1)/2

In this paper we investigate mixed sums of squares and triangular numbers. By means of q-series, we prove that any natural number n can be written as x + (2y) + Tz with x, y, z ∈ Z and Tz = z(z + 1)/2, this is stronger than a conjecture of Chen. Also, we can express n in any of the following forms: x + 2y + Tz , x 2 + 2y + 2Tz , x 2 + 2y + 4Tz , x 2 + 4y + 2Tz , 2x + 2y + Tz , x 2 + 2Ty + 2Tz , x 2 + 4Ty + Tz , x 2 + 4Ty + 2Tz , 2x + Ty + Tz , 2x 2 + 2Ty + Tz , 2x 2 + 4Ty + Tz , Tx + 4Ty + Tz , 2Tx + 2Ty + Tz , 2Tx + 4Ty + Tz . Concerning the converse we establish several theorems and make some conjectures.