Can you detect palpitations or arrhythmia when taking pulse?

I've been having occasional palpitations and a strange but mild sensation in my chest (not pain). When I check my pulse during one of these, I cannot detect an irregular rhythm. It seems like my heart is beating with evenly-spaced beats. Some of the beats may be harder than others, but everything seems regular. My pulse rate from a pulse oximeter seems normal and my BP seems normal.

For some reason though I can't shake the sensation that my heart is fluttering. Would I be able to detect this by checking my pulse?

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Yes, you can feel an irregular heart beat in your pulse but you probably shouldn’t constantly be checking your pulse as it will just cause your more anxiety.
Don’t forget your stomach, esophagus etc is all in the same vicinity and so it may very well be that whatever you’re feeling is coming from one of those and not you’re heart.