Liber Abaci is not associated with Greek arithmetic in Euclid's style, but with the Greek logistic tradition, of which we have not sufficient evidence (see the reference of Plato to it).

In addition to the observations by @Ioannis M. Vandoulakis on Liber Abaci and Greek arithmetic, it should be observed that the Fibonacci sequence is closely related to the golden ratio, which dates  back to Euclid's Elements.    The Liber Abaci, Fibonacci sequence and golden ratio are thoroughly examined in

N. Kowalsky, Mathematics in Nature, 2012:

http://web1.desales.edu/assets/salesian/PDF/HP2012-Kowalsky.pdf

Ioannis' and James' observations discuss related but not core scaled Greek arithmetic issues that included both theoretical and practical (logistica) arithmetic. Ioannis suggests only logistica was carried forward by Arabs, a point I do not see. James mentions the Golden mean, as aspect of theoretical arithmetic, a point is historical true, but not confirmed in the manner discussed. 

Arab numeration scaled rational numbers n/p subtracted LCM 1/m that modified Greek scaled rational numbers in one major respect, while retaining both sides of Greek theoretiucal and practical arithmetic in other applications. Greek rational numbers n/p were scaled by LCM m/m in a multiplication context such that mn/mp that was both theoretical (number theory) and practical (logistica) per : http://planetmath.org/arabicnumerals

As a followup to the observations by @Milo, consider first a detailed account of liber abaci from an ancient Arab mathematician's perspective is given in

J.D. Rey, Algebraic operations in an early sixteenth-century Catalan manuscript, 2008: 

http://www.upf.edu/hciencia/docampo4.pdf

And in an effort to get closer to an answer to the question for this thread, consider

J. Hoyrup, Proportions in Liber Abaci, 2008:

http://dspace.ruc.dk/bitstream/1800/13110/1/Hoyrup_2009_d___Proportions__in_the_Liber_abbaci.PDF

See, especially, the comments about Fibonacci on the 10 non-arithmetical means between two numbers studied in ancient Greek mathematics (p. 9) and the Table (p. 10) that compares various ratios relative to Pappos, Nicomachus and Liber Abaci.

@James, you cited reference paper mention a Liber Abaci point via Eves, my history of math professors' text. I know Eves' work well. I love Eves' views. Given that your cited paper was published in 2012 L.E.Sigler 's complete" Liber Abac"i translation, the first one in English,  should have also been stressed, in which three arithmetic notations connected Greeks like Euclid with ring numbers as well as re-scalng Greek multiplication  rational numbers to Arab rational numbers scaled by subtraction. Greek, Arab and Fibonacci's arithmetic relied on unit fractions filled with arithmoi and logistica fragments.to record rational numbers in all classes of problems.

@Milo, many thanks for pointing out L.E. Sigler's work.   See, for example:

file:///Users/jfpeters/Downloads/productFlyer_978-0-387-40737-1.pdf

Here is a review by A.F. Horadam of L.E. Sigler's translation of Leonardo Pisano's book of calculation:

http://www.fq.math.ca/Papers1/42-1/quarthoradam04review.pdf

This review contains a detailed section on Fibonacci fractions.    Also, Horadam gives high praise for Sigler's Introduction (pages 1-11 in Sigler's translation), which gives an account of Fibonacci's life and achievements.    Another detailed review of Sigler's translation is given by S. Cuomo,  Imperial College, University of London:

http://www.ircps.org/sites/ircps.org/files/aestimatio/1/2004-02-01_Cuomo.pdf

I also found another article by J. Hoyrup, which gives a very detailed look at Fibonacci's work on fractions:

http://smf4.emath.fr/Publications/RevueHistoireMath/11/pdf/smf_rhm_11_23-56.pdf

@James, thank you for stressing L.E. Sigler's 2002 publication of the 800 year old "Liber Abaci" . As one contrary view, JJ Sylvester suggested in 1891 that Fibonacci's 7th distinction represented a n-step  step greedy algorithm. Sigler's 2002 translation vividly  shows that only a  two-step method was at work per http://planetmath.org/liberabaci

That is, the historical Fibonacci math views have been modernized to fit the personal needs of modern mathematicians, JJ Sylvester being one., I'll not speak to your secondary references.  All are interesting. However, my view is that primary  references and raw data should precede secondary conclusions

Markovic,

The "Liber Abaci" took 1/3 of its text to define seven  methods (distinctions per LE Siglers' 2001 translation) that converted rational number n/p by subtracting LCM 1/m to not so concise unit fraction series. Most often 2-term series were obtained  per (n/p - 1/m) = (mn - p)/mp by setting (mn- p) = 1. When impossible, like (4/13 - 1/4) = (16 - 13)/52 = 3/52 a second LCM 1/18 calculated a 3-term series by the 7th method (distinction_ per (3/52 - 1/18) = (54 - 52)/968 = 1/468 meant 4/13 = 1/4 + 1/18 + 1/468, written in the reverse direction as Greeks and Egyptians scaled rational numbers by multiplication 4/13(4/4) = 16/52 = (13 + 2 + 1)/52 = 1/4 + 1/26 + 1/52, a more concise series.

In passing, Ahmes, an Egyptian in 1650 BCE took 1/3 of his text to convert 2/n , from 2/3 to 2/101, to concise 2-term, 3-term, 4-term and 5-term unit fraction series by selecting the best LCM m/m.

Bonjour,

J'ai peu d'informations sur le sujet, mais puis-je vous écrire en français?

Amand, per your question, Google translate allows anyone to write in their native languages per" I have little information on the subject, but can I write you in French.", please add your points on this topic.

James,,

Thank you for citing an Australian scholar that did not have the math background to confront Sigler's  translation issues. I also read Jens Hoyrup review, who did confront a few of Sigler translation issues.

 I know Hoyrup's math back ground well. About 20 years ago he reviewed one of my early Egyptian math papers, related to a submission to "Histsoria Mathematica" on an Egyptian fractions article..Hoyrup has been a Babylonian advocate for many years, and reported hints of Greek and Egyptian unit fraction fractions in reviewing Leonardo's  proportions, but insufficient to be of probative value. 

Rather than discussing Hoyrup's emerging  and incomplete views on medieval, Greek and Egyptian rational numbers represented by unit fraction series, may I discuss what  he said, and did not say, about Leonardo's proportions. Hoyrup took most of Sigler's proportion comments as valid. Leonardo's  proportion discussions, reviewed by Hoyrup consistently supported Sigler and indirectly stressed long time views that Babylonian arithmetic was superior to Egyptian arithmetic (by ignoring Egyptian fraction details) .:

Reading any secondary source paper like Sigler, care should be taken from the very beginning. Studying Leonardo's scaling of rational numbers by subtraction, a method that dominated 1/3 of the Liber Abaci, Sigler and Hoyrup, missed fundamental Arab notations foundations that modified Greek multiplication scaled rational numbers, and much more. For example, Hoyrup also missed an important inverse proportion reported in Leonardo's square root of 10. Leonard in his second step divided error1 = 1/36 by 2(3 + 1/6)  = 1/36 x 6/38 = 1/228, facts that were not discussed or footnoted by Sigler, either.

Leonardo clearly discussed the square root of 6 and square root of 10, the later stressed in an update of two of my 10 year old papers:

https://www.academia.edu/11810863/Liber_Abaci_2015_update

Best Regards,

Milo Gardner