Fibonacci sequence (English Wikipedia)

André-Jeannin, Richard (1989), "Irrationalité de la somme des inverses de certaines suites récurrentes", Comptes Rendus de l'Académie des Sciences, Série I, 308 (19): 539–41, MR 0999451
Williams, H. C. (1982), "A note on the Fibonacci quotient $F_{p-\varepsilon }/p$ ", Canadian Mathematical Bulletin, 25 (3): 366–70, doi:10.4153/CMB-1982-053-0, hdl:10338.dmlcz/137492, MR 0668957. Williams calls this property "well known".
Diaconis, Persi (2018), "Probabilizing Fibonacci numbers" (PDF), in Butler, Steve; Cooper, Joshua; Hurlbert, Glenn (eds.), Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, pp. 1–12, ISBN 978-1-107-15398-1, MR 3821829, archived from the original (PDF) on 2023-11-18, retrieved 2022-11-23
Cohn, J. H. E. (1964), "On square Fibonacci numbers", The Journal of the London Mathematical Society, 39: 537–540, doi:10.1112/jlms/s1-39.1.537, MR 0163867
Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers", Rendiconti del Circolo Matematico di Palermo, 49 (2): 313–18, doi:10.1007/BF02904236, ISSN 1973-4409, MR 1765401, S2CID 121789033
Broughan, Kevin A.; González, Marcos J.; Lewis, Ryan H.; Luca, Florian; Mejía Huguet, V. Janitzio; Togbé, Alain (2011), "There are no multiply-perfect Fibonacci numbers", Integers, 11a: A7, MR 2988067
Luca, Florian; Mejía Huguet, V. Janitzio (2010), "On Perfect numbers which are ratios of two Fibonacci numbers", Annales Mathematicae at Informaticae, 37: 107–24, ISSN 1787-6117, MR 2753031
Lü, Kebo; Wang, Jun (2006), " $k$ -step Fibonacci sequence modulo $m$ ", Utilitas Mathematica, 71: 169–177, MR 2278830
Stephenson, Kenneth (2005), Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, ISBN 978-0-521-82356-2, MR 2131318; see especially Lemma 8.2 (Ring Lemma), pp. 73–74, and Appendix B, The Ring Lemma, pp. 318–321.

archive.org

Lucas 1891, p. 3. Lucas, Édouard (1891), Théorie des nombres (in French), vol. 1, Paris: Gauthier-Villars.
Lucas 1891, p. 4. Lucas, Édouard (1891), Théorie des nombres (in French), vol. 1, Paris: Gauthier-Villars.
Lucas 1891, p. 7. Lucas, Édouard (1891), Théorie des nombres (in French), vol. 1, Paris: Gauthier-Villars.
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arxiv.org

Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969, S2CID 10266596

books.google.com

Goonatilake, Susantha (1998), Toward a Global Science, Indiana University Press, p. 126, ISBN 978-0-253-33388-9
Knuth, Donald (2006), The Art of Computer Programming, vol. 4. Generating All Trees – History of Combinatorial Generation, Addison–Wesley, p. 50, ISBN 978-0-321-33570-8, it was natural to consider the set of all sequences of [L] and [S] that have exactly m beats. ... there are exactly Fm+1 of them. For example the 21 sequences when $m = 7$ are: [gives list]. In this way Indian prosodists were led to discover the Fibonacci sequence, as we have observed in Section 1.2.8 (from v.1)
Knuth, Donald (1968), The Art of Computer Programming, vol. 1, Addison Wesley, p. 100, ISBN 978-81-7758-754-8, Before Fibonacci wrote his work, the sequence Fn had already been discussed by Indian scholars, who had long been interested in rhythmic patterns ... both Gopala (before 1135 AD) and Hemachandra (c. 1150) mentioned the numbers 1,2,3,5,8,13,21 explicitly [see P. Singh Historia Math 12 (1985) 229–44]" p. 100 (3d ed) ...
Livio 2003, p. 197. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, p. 197–198. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Sarah-Marie Belcastro (2018). Discrete Mathematics with Ducks (2nd, illustrated ed.). CRC Press. p. 260. ISBN 978-1-351-68369-2. Extract of page 260
Honsberger, Ross (1985), "Millin's series", Mathematical Gems III, Dolciani Mathematical Expositions, vol. 9, American Mathematical Society, pp. 135–136, ISBN 9781470457181
Stephenson, Kenneth (2005), Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, ISBN 978-0-521-82356-2, MR 2131318; see especially Lemma 8.2 (Ring Lemma), pp. 73–74, and Appendix B, The Ring Lemma, pp. 318–321.
Livio 2003, p. 110. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, pp. 112–13. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, p. 112. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, pp. 98–99. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, p. 176. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3
Livio 2003, p. 193. Livio, Mario (2003) [2002], The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (First trade paperback ed.), New York City: Broadway Books, ISBN 0-7679-0816-3

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colgate.edu

math.colgate.edu

Broughan, Kevin A.; González, Marcos J.; Lewis, Ryan H.; Luca, Florian; Mejía Huguet, V. Janitzio; Togbé, Alain (2011), "There are no multiply-perfect Fibonacci numbers", Integers, 11a: A7, MR 2988067

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Singh, Parmanand (1985), "The So-called Fibonacci numbers in ancient and medieval India", Historia Mathematica, 12 (3): 229–44, doi:10.1016/0315-0860(85)90021-7
Singh, Parmanand (1985), "The So-called Fibonacci Numbers in Ancient and Medieval India", Historia Mathematica, 12 (3), Academic Press: 232, doi:10.1016/0315-0860(85)90021-7
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Glaister, P (1995), "Fibonacci power series", The Mathematical Gazette, 79 (486): 521–25, doi:10.2307/3618079, JSTOR 3618079, S2CID 116536130
Williams, H. C. (1982), "A note on the Fibonacci quotient $F_{p-\varepsilon }/p$ ", Canadian Mathematical Bulletin, 25 (3): 366–70, doi:10.4153/CMB-1982-053-0, hdl:10338.dmlcz/137492, MR 0668957. Williams calls this property "well known".
Cohn, J. H. E. (1964), "On square Fibonacci numbers", The Journal of the London Mathematical Society, 39: 537–540, doi:10.1112/jlms/s1-39.1.537, MR 0163867
Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969, S2CID 10266596
Luo, Ming (1989), "On triangular Fibonacci numbers" (PDF), Fibonacci Quart., 27 (2): 98–108, doi:10.1080/00150517.1989.12429576
Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers", Rendiconti del Circolo Matematico di Palermo, 49 (2): 313–18, doi:10.1007/BF02904236, ISSN 1973-4409, MR 1765401, S2CID 121789033
Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–79, doi:10.2307/2325076, JSTOR 2325076
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Vogel, Helmut (1979), "A better way to construct the sunflower head", Mathematical Biosciences, 44 (3–4): 179–89, doi:10.1016/0025-5564(79)90080-4
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Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers", IEEE Transactions on Instrumentation and Measurement, IM-34 (4): 650–653, Bibcode:1985ITIM...34..650P, doi:10.1109/tim.1985.4315428, S2CID 35413237
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, hdl:11250/180781, S2CID 8550726

ektf.hu

ami.ektf.hu

Luca, Florian; Mejía Huguet, V. Janitzio (2010), "On Perfect numbers which are ratios of two Fibonacci numbers", Annales Mathematicae at Informaticae, 37: 107–24, ISSN 1787-6117, MR 2753031

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handle.net

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Williams, H. C. (1982), "A note on the Fibonacci quotient $F_{p-\varepsilon }/p$ ", Canadian Mathematical Bulletin, 25 (3): 366–70, doi:10.4153/CMB-1982-053-0, hdl:10338.dmlcz/137492, MR 0668957. Williams calls this property "well known".
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, hdl:11250/180781, S2CID 8550726

harvard.edu

ui.adsabs.harvard.edu

Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969, S2CID 10266596
Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers", IEEE Transactions on Instrumentation and Measurement, IM-34 (4): 650–653, Bibcode:1985ITIM...34..650P, doi:10.1109/tim.1985.4315428, S2CID 35413237

hmc.edu

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Su, Francis E (2000), "Fibonacci GCD's, please", Mudd Math Fun Facts, et al, HMC, archived from the original on 2009-12-14, retrieved 2007-02-23

jstor.org

Glaister, P (1995), "Fibonacci power series", The Mathematical Gazette, 79 (486): 521–25, doi:10.2307/3618079, JSTOR 3618079, S2CID 116536130
Freyd, Peter; Brown, Kevin S. (1993), "Problems and Solutions: Solutions: E3410", The American Mathematical Monthly, 99 (3): 278–79, doi:10.2307/2325076, JSTOR 2325076
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mersennus.net

Fibonacci and Lucas factorizations, Mersennus collects all known factors of $F (i)$ with $i < 10000$ .

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Varenne, Franck (2010), Formaliser le vivant - Lois, Théories, Modèles (in French), Hermann, p. 28, ISBN 9782705678128, retrieved 2022-10-30, En 1830, K. F. Schimper et A. Braun [...]. Ils montraient que si l'on représente cet angle de divergence par une fraction reflétant le nombre de tours par feuille ([...]), on tombe régulièrement sur un des nombres de la suite de Fibonacci pour le numérateur [...].

oeis.org

Sloane, N. J. A. (ed.), "Sequence A002390 (Decimal expansion of natural logarithm of golden ratio)", The On-Line Encyclopedia of Integer Sequences, OEIS Foundation
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redgolpe.com

fibonacci.redgolpe.com

Factors of Fibonacci and Lucas numbers, Red golpe collects all known factors of $F (i)$ with $10000 < i < 50000$ .

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Bugeaud, Y; Mignotte, M; Siksek, S (2006), "Classical and modular approaches to exponential Diophantine equations. I. Fibonacci and Lucas perfect powers", Ann. Math., 2 (163): 969–1018, arXiv:math/0403046, Bibcode:2004math......3046B, doi:10.4007/annals.2006.163.969, S2CID 10266596
Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers", Rendiconti del Circolo Matematico di Palermo, 49 (2): 313–18, doi:10.1007/BF02904236, ISSN 1973-4409, MR 1765401, S2CID 121789033
Patranabis, D.; Dana, S. K. (December 1985), "Single-shunt fault diagnosis through terminal attenuation measurement and using Fibonacci numbers", IEEE Transactions on Instrumentation and Measurement, IM-34 (4): 650–653, Bibcode:1985ITIM...34..650P, doi:10.1109/tim.1985.4315428, S2CID 35413237
Brasch, T. von; Byström, J.; Lystad, L.P. (2012), "Optimal Control and the Fibonacci Sequence", Journal of Optimization Theory and Applications, 154 (3): 857–78, doi:10.1007/s10957-012-0061-2, hdl:11250/180781, S2CID 8550726

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Diaconis, Persi (2018), "Probabilizing Fibonacci numbers" (PDF), in Butler, Steve; Cooper, Joshua; Hurlbert, Glenn (eds.), Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, pp. 1–12, ISBN 978-1-107-15398-1, MR 3821829, archived from the original (PDF) on 2023-11-18, retrieved 2022-11-23

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Su, Francis E (2000), "Fibonacci GCD's, please", Mudd Math Fun Facts, et al, HMC, archived from the original on 2009-12-14, retrieved 2007-02-23
Diaconis, Persi (2018), "Probabilizing Fibonacci numbers" (PDF), in Butler, Steve; Cooper, Joshua; Hurlbert, Glenn (eds.), Connections in Discrete Mathematics: A Celebration of the Work of Ron Graham, Cambridge University Press, pp. 1–12, ISBN 978-1-107-15398-1, MR 3821829, archived from the original (PDF) on 2023-11-18, retrieved 2022-11-23
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Hutchison, Luke (September 2004), "Growing the Family Tree: The Power of DNA in Reconstructing Family Relationships" (PDF), Proceedings of the First Symposium on Bioinformatics and Biotechnology (BIOT-04), archived from the original (PDF) on 2020-09-25, retrieved 2016-09-03

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Luca, Florian (2000), "Perfect Fibonacci and Lucas numbers", Rendiconti del Circolo Matematico di Palermo, 49 (2): 313–18, doi:10.1007/BF02904236, ISSN 1973-4409, MR 1765401, S2CID 121789033
Luca, Florian; Mejía Huguet, V. Janitzio (2010), "On Perfect numbers which are ratios of two Fibonacci numbers", Annales Mathematicae at Informaticae, 37: 107–24, ISSN 1787-6117, MR 2753031

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