A result similar to Lagrange's theorem

Abstract: Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for $35$ triples $(b,c,d)$ with $1\le b\le c\le d$ (including $(2,3,4)$ and $(2,4,8)$), we prove that any nonnegative integer can be exprssed as $p_8(w)+bp_8(x)+cp_8(y)+dp_8(z)$ with $w,x,y,z\in\mathbb Z$. We also pose several conjectures for further research.