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

To find a recurrence relation for the given sequence (4, 14, 44, 134, 404, ...), let's examine the pattern in the terms. Notice that __________ __________ _________ ____ ______ ______ _____.
____ ________ _______ ________ ________ _______ __________ _____ _________ _____ _______ ________.
____ __________ ______ ______ _____ _______ _____ ____ ________ ___.
___ __________ ___ ________ _____ ______ ____ ______ _________ ________ ______ ___.
________ ___ _______ ________ _________ _______ ___.
_____ ___ _____ ________ _______ __________ ___ _______.
_________ ______ _____ ____ _____.
________ __________ ___ __________ __________ _____ _____ _________.
_____ _________ ________ ______ ____ __________ ___ __________ ______ ______.
________ __________ ________ _________ _______ ____ ___.
_______ ____ ______ _____ __________ ___ ________ ______.
__________ ___ _______ ___ ___ ______ _____ ________.
_________ ________ ___ _____ __________ ______ ________ _________ ______ _________.
_____ _________ __________ ____ ___ ______ ___ ______ _____ _________.
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