Find a recurrence relation and initial conditions for 4, 14, 44, 134, 404, …

To find a recurrence relation and initial conditions for the given sequence \(4, 14, 44, 134, 404, \ldots\), let's analyze the pattern.

Observing the sequence, we can see that each term is obtained by _____ ______ _____ _________ ___ ___.
__________ ____ ______ ________ ___ _____ ________ _____.
____ ____ ____ _______ _________ ______.
______ ____ _________ ___ _____ __________ ____ _____.
______ ____ _____ _______ __________ __________.
_____ ___ ________ _________ _______ ______ ______ ____ _________ ____.
__________ ________ _______ _________ _________ ____ ________ _______.
________ _______ ___ _____ ____ _____ ________ ________.
___ ____ ________ _________ __________ ______ ____ ______ _______ ______.
_________ ________ _______ __________ _________ ________ __________ __________ _____ _______.
______ _______ __________ ________ __________ _______ _______ _____ ___.
______ ________ _________ _________ __________.
___ _________ _____ _________ __________ ________.
_______ __________ ________ ________ _________ _______.
_______ ________ _________ _____ _________ _______ _________ __________ ________ ___ ____ _______.
_________ ___ ____ __________ ________ ______ _____ ___ _______ ___.
__________ ____ ______ _________ ____ ___ _____ ___.
________ ____ ______ _________ _______ __________ ____ __________ __________ ___ ________.
______ ___ _______ _____ ______ ____ _____.
__________ _________ __________ ___ ______ ______ ____ ___ _____ ___.
_________ ______ _______ _______ _____ ___ __________ _____ ____ ____.
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