It's (that is, the protons) neighbouring carbon does not have any protons with which it can magnetically interact, and so no splitting occurs.
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Remember, the structure is CH3CH2CH2-C-H
The carbon atom adjacent to the hydrogen in question, has no hydrogen atoms bonde to it. So no splitting occurs.
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When we talk of splitting on protons of the adjacent carbons, we mean adjacent to the proton in question (not the carbon to which it is attached).
Hydrogens that couple (split each other) are considered as two bonds away, three bonds away (and sometimes for very weak splitting) four bonds away.
In this compound, the aldehyde H is three bonds away from the hydrogens on the -CH2- (ie H-CO-C-H (three hyphens representing three bonds).
No splitting occurs - they are not neighbouring (two bonds away from each other) and magnetic interactions aren't typically carried across a carbon atom).