A result similar to Lagrange's theorem

Generalized octagonal numbers are those p8(x)=x(3x−2) with x∈Z. In this paper we 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⩽b⩽c⩽d (including (2,3,4) and (2,4,8)), we prove that any nonnegative integer can be expressed as p8(w)+bp8(x)+cp8(y)+dp8(z) with w,x,y,z∈Z. We also pose several conjectures for further research.