The graph of $y=ax^2+bx+c$ is given below, where $a$, $b$, and $c$ are integers.  Find $a+b+c$.

[asy]
Label f;

f.p=fontsize(4);

xaxis(-3,3,Ticks(f, 1.0));

yaxis(-4,4,Ticks(f, 1.0));

real f(real x)

{

return -2x^2+4x+1;

}

draw(graph(f,-.7,2.7),linewidth(1),Arrows(6));
[/asy]
When $x=1$ we have $y=a+b+c$.  Since $a$, $b$, and $c$ are integers, we know that $y$ must be an integer when $x=1$.  The graph passes through $(1,3)$, so $y=3$ when $x=1$, which means $a+b+c=\boxed{3}$.