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"We thank Daniele Dorigoni for identifying this function using The On-Line Encyclopedia of Integer Sequences." [Joseph A. Farrow and Arthur E. Lipstein, 2017]

"The On-Line Encyclopedia of Integer Sequences (OEIS) is a triumph, residing at the intersection of mathematics and technology, and appealing to specialists and novices alike. It serves not just as an immense repository (to inform authors of data and connections they may have overlooked), but also as a feedback loop (to motivate and drive research into previously unimagined lines of thought)." [Steven Finch, 2021]

"We would like to acknowledge the mathematical software Sage [44] and the Online Encyclopedia of Integer Sequences [33] which were both essential for our investigations." [Laura Florescu et al., 2015]

"Sloane's influential On-Line Encyclopedia of Integer Sequences is an indispensable research tool in the service of the mathematical community..." [A. S. Fraenkel, 2010]

"The connection between tensor products of A_oo-maps and hypercubes (Remark 4.2) was discovered by consulting the OEIS." [Matthias Franz, 2015]


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References

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  2. V. Fack, S. Lievens and J. Van der Jeugt, On rotation distance between binary coupling trees and applications for 3nj-coefficients, Comput. Phys. Commun., 119 (1999) 99-114.
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  4. Vinicius Facó, D Marques, Tribonacci Numbers and the Brocard-Ramanujan Equation, - Journal of Integer Sequences, Vol. 19, 2016, #16.4.4.
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  6. Barry Fagin, Search Heuristics and Constructive Algorithms for Maximally Idempotent Integers, Information (2021) Vol. 12, No. 8, 305. doi:10.3390/info12080305 (A306508)
  7. P. Fahr, Infinite Gabriel-Roiter measures for the 3-Kronecker quiver, PhD. U. Bielfeld (2008)
  8. P. Fahr, C. M. Ringel, A partition number for Fibonacci Numbers, JIS 11 (2008) 08.1.4.
  9. Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, JIS 15 (2012) 12.2.1.
  10. Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019. See also Relational and Algebraic Methods in Computer Science, RAMiCS: International Conference on Relational and Algebraic Methods in Computer Science, Lecture Notes in Computer Science (LNCS, Vol. 12062) Theoretical Computer Science and General Issues (LNTCS, Vol. 12062), Springer, Cham, 82-99. doi:10.1007/978-3-030-43520-2_6 (A000112, A003430, A079566)
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  20. S. Falcon, On the Lucas triangle and its relationship with the k-Lucas numbers, Journal of Mathematical and Computational Science, 2 (2012), No. 3, 425-434. Available online at http://scik.org.
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  22. Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; doi:10.4134/CKMS.2013.28.4.827.
  23. S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; doi:10.4236/am.2014.515216
  24. S. Falcon, On The Generating Functions of the Powers of the K-Fibonacci Numbers, Scholars Journal of Engineering and Technology (SJET), 2014; 2 (4C):669-675; http://saspublisher.com/wp-content/uploads/2014/06/SJET24C669-675.pdf
  25. S. Falcon, On the k-Jacobsthal Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 67-77, ISSN 2374-2348 (Print) 2374-2356 (Online); http://aripd.org/journals/arms/Vol_2_No_1_March_2014/8.pdf
  26. S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
  27. S. Falcon, On the Sequences of Products of Two k-Fibonacci Numbers, American Review of Mathematics and Statistics, March 2014, Vol. 2, No. 1, pp. 111-120.
  28. S. Falcon, Generalized (k, r)–Fibonacci Numbers, Gen. Math. Notes, Vol. 25, No. 2, December 2014, pp.148-158; http://www.geman.in
  29. Sergio Falcon, The k–Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153–157.
  30. Sergio Falcon, On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944; doi:10.1080/23311835.2016.1201944
  31. Sergio Falcon, <a href="http://ijism.org/administrator/components/com_jresearch/files/publications/IJISM_816_FINAL.pdf">Generating Function of Some k-Fibonacci and k-Lucas Sequences</a>, International Journal of Innovation in Science and Mathematics (2019) Vol. 7, Issue 2, 2347–9051.
  32. Sergio Falcón, Binomial Transform of the Generalized k-Fibonacci Numbers, Communications in Mathematics and Applications (2019) Vol. 10, No. 3, 643–651. doi:10.26713/cma.v10i3.1221 (A000045, A000129, A000931, A002605, A015518, A052921, A114199, A114203, A312448)
  33. Sergio Falcon, Angel Plaza, On k-Fibonacci numbers of arithmetic indexes, Applied Mathematics and Computation, Volume 208, Issue 1, 1 February 2009, Pages 180-185.
  34. Sergio Falcon, Angel Plaza, On k-Fibonacci sequences and polynomials and their derivatives, Chaos, Solitons & Fractals, Volume 39, Issue 3, 15 February 2009, Pages 1005-1019.
  35. Sergio Falcon, Angel Plaza, The metallic ratios as limits of complex valued transformations, Chaos, Solitons & Fractals, Volume 41, Issue 1, 15 July 2009, Pages 1-13.
  36. Sergio Falcon, Angel Plaza, k-Fibonacci sequences modulo m, Chaos, Solitons & Fractals, Volume 41, Issue 1, 15 July 2009, Pages 497-504.
  37. Sergio Falcon, Half self-convolution of the k-Fibonacci sequence, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 3, 96–106. doi:10.7546/nntdm.2020.26.3.96-106 (A001629, A006645, A014334, A084137, A099924, A203578, A203579)
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  39. Joshua Fallon, Kirsten Hogenson, Lauren Keough, Mario Lomelí, Marcus Schaefer, Pablo Soberón, A Note on the Maximum Rectilinear Crossing Number of Spiders, arXiv:1808.00385 [math.CO], 2018. (A248380)
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  41. Justine Falque, On the enumeration of P-oligomorphic groups, Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020), hal-02918958 [math.cs], 25-26. Abstract (A000638)
  42. Justine Falque, Jean-Christophe Novelli, and Jean-Yves Thibon, Pinnacle sets revisited, arXiv:2106.05248 [math.CO], 2021. (A001045, A007477)
  43. Bruce Fang, Pamela E. Harris, Brian M. Kamau, and David Wang, Vacillating parking functions, arXiv:2402.02538 [math.CO], 2024. (A001333, A040000)
  44. Jin-Hui Fang and Jie Ma, The sum of reciprocals of least common multiples, Period. Math. Hung. (2021). doi:10.1007/s10998-021-00395-w
  45. Jonathan Fang, Zachary Hamaker, Justin Troyka, On pattern avoidance in matchings and involutions, arXiv:2009.00079 [math.CO], 2020. (A084261, A122852)
  46. Qi Fang, Ya-Nan Feng, and Shi-Mei Ma, Alternating runs of permutations and the central factorial numbers, arXiv:2202.13978 [math.CO], 2022. (A008971, A036969, A160562)
  47. Wenjie Fang, A partial order on Motzkin paths, Discrete Math., 343 (2020), #111802.
  48. Wenjie Fang, Hsien-Kuei Hwang, Mihyun Kang, Phase transitions from exp(n1/2) to exp(n2/3) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020. (A000041, A000219, A003293, A155585, A266648)
  49. Xin Fang and Ghislain Fourier, Torus fixed points in Schubert varieties and Genocchi numbers, arXiv:1504.03980.
  50. Reza Farhadian, A New Conjecture On the primes, Preprint, 2016; http://www.primepuzzles.net/conjectures/Reza%20Faradian%20Conjecture.pdf
  51. Reza Farhadian, Rafael Jakimczuk, A Note on the Asymptotic Behavior of the Distribution Function of a General Sequence, Annales Mathematicae Silesianae (2021) Vol. 35, Issue 1, 44-54. doi:10.2478/amsil-2020-0027 (A000110, A000142, A002110)
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  53. Bakir Farhi, Summation of Certain Infinite Lucas-Related Series, J. Int. Seq., Vol. 22 (2019), Article 19.1.6. HTML (A000032, A000045, A000051, A000129, A000225, A002203)
  54. Bakir Farhi, Some Applications of the Lagrange Inversion Formula for the k-Fibonacci Numbers, J. Int. Seq (2024) Vol. 27, Art. 24.1.1. Abstract (A000045)
  55. Sara Faridi, Mohammad Farrokhi Derakhshandeh Ghouchan, Roghayyeh Ghorbani, and Ali Akbar Yazdan Pour, Cellular resolutions of monomial ideals and their Artinian reductions, arXiv:2209.10338 [math.AC], 2022.
  56. Michael Farina and Armando Grez, New Upper Bounds on the Distance Domination Numbers of Grids, Rose-Hulman Undergraduate Mathematics Journal, Volume 17, No. 2, Article 7, Fall 2016.
  57. G. Farkas, G. Kallos and G. Kiss, Large primes in generalized Pascal triangles Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171; http://www.acta.sapientia.ro/acta-info/C3-2/info32-2.pdf.
  58. Elin Farnell, Puzzle Pedagogy: A Use of Riddles in Mathematics Education, PRIMUS, July 2016, pp. 202-211; doi:10.1080/10511970.2016.1195465
  59. M. Farrokhi D. G., "Some Remarks On the Equation Fn = kFm In Fibonacci Numbers", J. Integer Sequences, Volume 10, 2007, Article 07.5.7.
  60. Joseph A. Farrow, A Monte Carlo Approach to the 4D Scattering Equations, arXiv:1806.02732 [hep-th], 2018. (A008292)
  61. Joseph A. Farrow and Arthur E. Lipstein, From 4d Ambitwistor Strings to On Shell Diagrams and Back, arXiv:1705.07087v1 [hep-th], 2017. (OEIS only cited in version 1) ["We thank Daniele Dorigoni for identifying this function using The On-Line Encyclopedia of Integer Sequences"]
  62. Farrugia, Alexander doi:10.1016/j.dam.2016.04.001 On strongly asymmetric and controllable primitive graphs. Discrete Appl. Math. 211, 58-67 (2016).
  63. Alexander Farrugia, The rank of Pseudo walk matrices: controllable and recalcitrant pairs, Open J. of Disc. Appl. Math. (2020) Vol. 3, No. 3, 41-52. doi:10.30538/psrp-odam2020.0042 (A001349)
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  67. Massimiliano Fasi, Jishe Feng, and Gian Maria Negri Porzio, Corrigendum to "Determinants of Normalized Bohemian Upper Hessenberg Matrices", The Electronic Journal of Linear Algebra (2021) Vol. 37, 160-162. doi:10.13001/ela.2021.5849 (A000051)
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  124. Margherita Maria Ferrari, Norma Zagaglia Salvi, Aperiodic Compositions and Classical Integer Sequences, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.8. PDF
  125. R. Ferraz de Andrade, Erik Lundberg, B. Nagle, Asymptotics of the extremal excedance set statistics, Eur. J. Comb. 46 (2015) 78-88 doi:10.1016/j.ejc.2014.11.008
  126. José William Porras Ferreira, An Approach to Solve Erdös-Straus Conjecture, ResearchGate, 2017.
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  131. L. Ferretti, F. Disanto and T. Wiehe, The Effect of Single Recombination Events on Coalescent Tree Height and Shape, PLoS ONE 8(4): e60123. doi:10.1371/journal.pone.0060123.
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  409. Robert Frontczak, Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, Vol. 12 (2018), No. 1, pp. 19-24. doi:10.12988/ijma.2018.712153 (A000073, A001644, A005277)
  410. Robert Frontczak, Convolutions for Generalized Tribonacci Numbers and Related Results, International Journal of Mathematical Analysis (2018) Vol. 12, Issue 7, 307-324. doi:10.12988/ijma.2018.8429 (A000073, A001644)
  411. Robert Frontczak, A Note on Hybrid Convolutions Involving Balancing and Lucas-Balancing Numbers, Applied Mathematical Sciences, Vol. 12, 2018, No. 25, 1201-1208. doi:10.12988/ams.2018.87111 (A001109, A001541)
  412. Robert Frontczak, Sums of Balancing and Lucas-Balancing Numbers with Binomial Coefficients, Int'l J. of Math. Analysis (2018) Vol. 12, No. 12, 585-594. PDF (A001109, A001541)
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  414. Robert Frontczak, Identities for generalized balancing numbers, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 2, 169-180. PDF (A000032, A000045, A001109, A001541)
  415. Robert Frontczak, Finite sums with generalized harmonic numbers. doi:10.13140/RG.2.2.28300.08320 (A000032, A000045)
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  419. Andrea Frosini, Giulia Palma, Elisa Pergola, Simone Rinaldi, A study on the fixed points of the γ function, arXiv:1911.04795 [math.CO], 2019.
  420. A. Frosini and S. Rinaldi, "On the Sequence A079500 and Its Combinatorial Interpretations", J. Integer Sequences, Volume 9, 2006, Article 06.3.1.
  421. Christiane Frougny, Edita Pelantová, and Milena Svobodová, "Minimal Digit Sets for Parallel Addition in Non-Standard Numeration Systems", Journal of Integer Sequences, Vol. 16 (2013), #13.2.17.
  422. Timothy J Frye, Recursive Relationships in the Classes of Odd Graphs and Middle Levels Graphs, arXiv preprint arXiv:1611.06401, 2016
  423. Amy M. Fu, A Context-free Grammar for Peaks and Double Descents of Permutations, arXiv:1801.04397 [math.CO], 2018. (A008971)
  424. Amy M. Fu, Frank Z.K. Li, Joint Distributions of Permutation Statistics and the Parabolic Cylinder Functions, arXiv:1809.07465 [math.CO], 2018. (A000085, A008971)
  425. Hao Fu, GN Han, Computer assisted proof for Apwenian sequences related to Hankel determinants, arXiv preprint arXiv:1601.04370, 2016
  426. H.-L. Fu, Y.-H. Lo and K. W. Shum, Optimal conflict-avoiding codes of odd length and weight three, Designs, Codes and Cryptography, Nov. 2012; doi:10.1007/s10623-012-9764-5.
  427. Shishuo Fu, Emma Yu Jin, Zhicong Lin, Sherry H.F. Yan, Robin D.P. Zhou, A new decomposition of ascent sequences and Euler-Stirling statistics, arXiv:1909.07277 [math.CO], 2019. (A022493)
  428. Shishuo Fu, Zhicong Lin, Yaling Wang, Refined Wilf-equivalences by Comtet statistics, arXiv:2009.04269 [math.CO], 2020. (A010683, A095264)
  429. Shishuo Fu, Zhicong Lin, Yaling Wang, A combinatorial bijection on di-sk trees, arXiv:2011.11302 [math.CO], 2020. (A009766)
  430. Shishuo Fu, Z. Lin, J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184, 2015
  431. Shishuo Fu, Zhicong Lin, Jiang Zeng, On two unimodal descent polynomials, Discrete Mathematics (2018), Vol. 341, Issue 9, 2616-2626. doi:10.1016/j.disc.2018.06.010
  432. Shishuo Fu and James Sellers, Enumeration of the degree sequences of line-Hamiltonian multigraphs, INTEGERS 12 (2012), #A24.
  433. Shishuo Fu, D Tang, Partitions with fixed largest hook length, arXiv preprint arXiv:1604.04028, 2016
  434. Shishuo Fu, Dazhao Tang, Generalizing a partition theorem of Andrews, arXiv:1705.05046 [math.CO], 2017. (A353902, A353903)
  435. Shishuo Fu, Dazhao Tang, Bin Han, Jiang Zeng, Gamma expansions of q-Narayana polynomials, pattern avoidance and the (−1)-phenomenon. arXiv:1805.08945 [math.CO], 2018. (A027307, A032349, A084868)
  436. Shishuo Fu, Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019. (A000007, A000012, A000079, A000108, A000244, A000957, A001003, A001700, A006318, A007318, A007564, A010054, A033184, A038207, A039598, A047891, A065600, A080247, A104219, A108524, A114709, A114710, A130595)
  437. Shishuo Fu, Yaling Wang, Bijective proofs of recurrences involving two Schröder triangles, European Journal of Combinatorics (2020) 103077. doi:10.1016/j.ejc.2019.103077
  438. Tung-Shan Fu and Chien-Tai Ting, Sign-balances of Tableaux with at Most Three Rows, Taiwanese J. Math. Advance Publication (2021) 1-19. doi:10.11650/tjm/210305 (A001006, A001405)
  439. Yixing Fu, E. J. König, J. H. Wilson, Yang-Zhi Chou, J. H. Pixley, Magic-angle semimetals, arXiv:1809.04604 [cond-mat.str-el], 2018. (A001076)
  440. Ekaterina Fuchkina, Sven Schneider, and Martin Bielik, Blocked All-Pairs Shortest Paths Dijkstra for Accelerating Grid-based Spatial Analysis, Proc. 13th Space Syntax Symp. (Weimar, Germany 2022). PDF (A054247)
  441. Christopher A. Fuchs, Maxim Olchanyi, and Matthew B. Weiss, Quantum mechanics? It's all fun and games until someone loses an i. arXiv:2206.15343 [quant-ph], 2022. (A002853)
  442. Michael Fuchs, Enumeration and Stochastic Properties of Tree-Child Networks, National Chengchi Univ. (Taipei 2023). PDF (A213863)
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  446. Y. Fujiwara, Parsing a Sequence of Qubits, IEEE Trans. Information Theory, 59 (2013), 6796-6806.
  447. Henryk Fukś, Three mathematical papers of Adam Adamandy Kochanski - annotated English translation, Antiq. math. 9 (2015) 31-65 doi:10.14708/am.v9i0.803
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  450. H. Fukś and J. M. G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv preprint arXiv:1306.1189, 2013
  451. Motohisa Fukuda, Ion Nechita, Enumerating meandric systems with large number of components, arXiv preprint arXiv:1609.02756, 2016.
  452. Damiano Fulghesu, James A. Sellers, and Courtney K. Taylor, Infinite Families of Infinite Series With Integer Sums, College Math. J. (2023). doi:10.1080/07468342.2022.2160611
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  454. Neale L. Fulton, Mark Westcott, Warren F. Smith, Constructing a Feasible Design Space for Multiple Cluster Conflict and Taskload Assessment, In: Sarker R., Abbass H., Dunstall S., Kilby P., Davis R., Young L. (eds) Data and Decision Sciences in Action, Lecture Notes in Management and Industrial Engineering. doi:10.1007/978-3-319-55914-8_15
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  456. Éric Fusy, Luca Lionni, Adrian Tanasa, Combinatorial study of graphs arising from the Sachdev-Ye-Kitaev model, arXiv:1810.02146 [math.CO], 2018. (A062980)

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