The Unknown Polynome

The condition $f(x+3) = f(x) - x^2 + x - 5$ gives a simple set of simultaneous equations for the coefficients of $f(x)$ . Since the degree of $f(x+3) - f(x)$ must be one less than the degree of $f(x)$ , it is clear that $f(x)$ must be cubic and solving the simultaneous equations tells us that $f(x) \; = \; -\tfrac19x^3 + \tfrac23x^2 - \tfrac83x + d$ for some constant $d$ . Since $f(1) = 3$ we deduce that $d = \tfrac{46}{9}$ . Hence $f(310) = \boxed{-3246866}$ .

$f(1)=3$ $f(1+3)=f(4)=f(1)-1^2+1-5$ $f(4+3)=f(7)=f(4)-4^2+4-5=f(1)-1^2+1-5-4^2+4-5$ $f(7+3)=f(10)=f(7)-7^2+7-5=f(1)-1^2+1-5-4^2+4-5-7^2+7-5$ $\cdots$ $f(307+3)=f(310)=f(307)-307^2+307-5=f(1)-1^2+1-5-4^2+4-5-7^2+7-5 \cdots -304^2+304-5-307^2+307-5$ $f(1)+\sum_{n=0}^{102} (-(3n+1)^2+(3n+1)-5)=-3246866$

I'm lazy. So, I got a computer to calculate it for me...

var y = 3;
var x = 1;

for (var i = 1; i <= 310; i += 3) {
  console.log(x, y)
  y = y - (x*x) + x - 5;
  x += 3;
}