Geodesic
The Mean Number of Steps in the Euclidean Algorithm with Odd Incomplete Quotients
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 594-604

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The length of the continued-fraction expansion of a rational number with odd incomplete quotients is expressed via the Gauss–Kuzmin statistics for the classical continued fraction. This has made it possible to prove asymptotic formulas, similar to those already known for the classical Euclidean algorithm, for the mean length of the Euclidean algorithm with odd incomplete quotients.
Keywords: Euclidean algorithm, Gauss–Kuzmin statistics, continued-fraction expansion, dual fraction, incomplete quotient. 
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     author = {A. V. Ustinov},
     title = {The {Mean} {Number} of {Steps} in the {Euclidean} {Algorithm} with {Odd} {Incomplete} {Quotients}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {594--604},
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     volume = {88},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a10/}
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A. V. Ustinov. The Mean Number of Steps in the Euclidean Algorithm with Odd Incomplete Quotients. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 594-604. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a10/