New results similar to Lagrange's four-square theorem

In this paper we establish some new results similar to Lagrange’s four-square theorem. For example, we prove that any integer $n>1$ can be written as $w(5w+1)\mathrm{/}2+x(5x+1)\mathrm{/}2+y(5y+1)\mathrm{/}2+z(5z+1)\mathrm{/}2$ with $w,x,y,z\in \mathbb{Z}$. Let $a$ and $b$ be integers with $a>0$, $b>-a$ and $\gcd (a,b)=1$. When $2\nmid ab$, we show that any sufficiently large integer can be written as $$\frac{w(aw+b)}{2}+\frac{x(ax+b)}{2}+\frac{y(ay+b)}{2}+\frac{z(az+b)}{2}$$ with $w,x,y,z$ nonnegative integers. When $2\mid a$ and $2\nmid b$, we prove that any sufficiently large integer can be written as $$w(aw+b)+x(ax+b)+y(ay+b)+z(az+b)$$ with $w,x,y,z$ nonnegative integers.