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In this paper we investigate the asymptotic behavior of the Riesz $s$-energyof the first $N$ points of a greedy $s$-energy sequence on the unit circle, forall values of $s$ in the range $0\leq s<\infty$ (identifying as usual the case$s=0$ with the logarithmic energy). In the context of the unit circle, greedy$s$-energy sequences coincide with the classical Leja sequences constructedusing the logarithmic potential. We obtain first-order and second-orderasymptotic results. The key idea is to express the Riesz $s$-energy of thefirst $N$ points of a greedy $s$-energy sequence in terms of the binaryrepresentation of $N$. Motivated by our results, we pose some conjectures forgeneral sequences on the unit circle.

Asymptotics of the energy of sections of greedy energy sequences on the unit circle, and some conjectures for general sequences