Loop in Inside Llewyn Davis

It's a deliberate choice by the Coen brothers:

There are the obvious ways the film kicks against traditional three-act movie structure. In a conversation with Guillermo del Toro featured in the extras of the Criterion package, the Coens assert that the idea of starting at the end and bringing us back there through the course of the film was one of the first concepts they thought of. Indeed, the story now goes that the image of a folk singer being beaten in an alley was the kernel of the whole film — grist to the mill of those who accuse the Coens (wrongly) of misanthropy. But even discounting time-manipulation formal trickery, the film’s story, told chronologically, is a circle, or at least the first complete circuit in an ever-decreasing spiral.

It’s a story that loops back on itself to deposit Llewyn right where we found him, just a little more broken. The terminally lovely Gorfeins have forgiven him; Ulysses the cat is back in the loving bosom of Lillian, who is making another of her "famous" ethnic dishes; Jean (Carey Mulligan) will revert to a state of not-pregnant with Llewyn’s maybe-child; and even the new knowledge Llewyn has gained will not make any measurable impact on his actual life. He will never go to Akron to see the child whose existence he’s been made aware of, and even his bid to change things up by abandoning his artistic dream and rejoining the Merchant marines comes to naught (Llewyn even fails at selling out). The world is conspiring to keep him exactly where he is.