Let´s take any x>3 and choose a,b such that a<b,a-b even and ab=x^2-1.
Then with y=(b-a)/2, z=(b+a)/2 we have x^2 + y^2 = z^2 + 1.
Particular cases∶
if x even then x^2 + ((x^2-2)/2)^2 = (x^2/2)^2 + 1; 
if x odd then x^2 + ((x^2-5)/4)^2 = ((x^2+3)/4)^2 + 1.