I think Paps knows Sans so well he knows exactly when his brother needs someone despite wanting to be left alone. When Sans needs a hug Paps gives him a hug. EDIT: Fixed wrong pictures order Dis is ART!
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An interesting idea. This would explain Sans feeling of responsibility to the underground to stop Frisk and be a good older brother. Sans would feel like he needed to be strong like his father and stand up to the bullies