Linear sampling Gaussian Blur

[UraniumSlug]

I've recently been experimenting with implementing a gaussian blur which samples half texels in attempt to retrieve the data of both texels. However I'm slightly skeptical of my results so I'm going to post my methods here and an output of the image.

I'm currently calculating my weights and offsets like so: weight1 * index + weight2 + (index + 1). After which I divide by the texture size.

[UraniumSlug]

When I use i as the offset and then multiply by

vec2 pixel_size = vec2(0.5 / tex_size.x, 0.5 / tex_size.y);

Would this make any difference?