The function rk(x) denotes the number of ways an integer x can be expressed as the sum of squares of k integers [the integers can be positive, can be negative, can be zero].
What is the value of r2(2013) + r2(2013-1) + r2(2013-4) + r2(2013-9) + r2(2013-16) + r2(2013-25) + ... + r2(2013-44^2)?
Note that 2013-45^2 is negative, so r2(2013-p^2) is always 0 if p>44.
My approach is as follows. A general term in the question is of the form r2(2013-k^2). It is the number of integer solutions of a^2 + b^2= 2013-k^2 -> a^2 + b^2 + k^2= 2013. So the answer should be r3(2013).
I searched for the function r3 but everywhere I saw complex calculations and use of calculus. Can anybody please give me a simple computation of r3(2013)?
Note:- WolframAlpha gives the answer 192, but I want a mathematical argument.