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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]

"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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  • This is part of the series of OEIS Wiki pages that list works citing the OEIS.
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  1. V. Fack, S. Fanscali, L. Szabolcs, L. Storme, G. van de Voorde, J. Winne, Small weight codewords in the codes arising from Desarguesian projective planes, Des. Codes Crypt. 46 (1) (2007) 25-43 doi:10.1007/s10623-007-9126-x
  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.
  3. V. Fack, S. Lievens and J. Van der Jeugt, On the diameter of the rotation graph of binary coupling trees, Discrete Mathematics 245 (2002) 1-18, doi:10.1016/S0012-365X(01)00418-6
  4. Vinicius Facó, D Marques, Tribonacci Numbers and the Brocard-Ramanujan Equation, - Journal of Integer Sequences, Vol. 19, 2016, #16.4.4.
  5. Barry Fagin, Idempotent Factorizations of Square-free Integers, Preprints (2019), also in Information (2019) Vol. 10, No. 7, 232. doi:10.3390/info10070232 (A306360, A306508, A306812, A318555)
  6. P. Fahr, Infinite Gabriel-Roiter measures for the 3-Kronecker quiver, PhD. U. Bielfeld (2008)
  7. P. Fahr, C. M. Ringel, A partition number for Fibonacci Numbers, JIS 11 (2008) 08.1.4.
  8. Philipp Fahr and Claus Michael Ringel, Categorification of the Fibonacci Numbers Using Representations of Quivers, JIS 15 (2012) 12.2.1.
  9. Uli Fahrenberg, Christian Johansen, Georg Struth, Ratan Bahadur Thapa, Generating Posets Beyond N, arXiv:1910.06162 [cs.FL], 2019. (A000112, A003430, A079566)
  10. N.-E. Fahssi, The polynomial triangles revisited, Arxiv preprint arXiv:1202.0228, 2012
  11. N.-E. Fahssi, A Systematic Study of Polynomial Triangles, Electronic Journal of Combinatorics, 2012.
  12. N.-E. Fahssi, The many aspects of polynomial triangles, Conference on Discrete Mathematics and Computer Science, Algeria, Sidi Bel Abbès, Nov 15-19, 2015; Recits Laboratory, Faculty of Mathematics, USTHB;; pages 147-150.
  13. Nour-Eddine Fahssi, On the combinatorics of exclusion in Haldane fractional statistics, arXiv:1808.00045 [cond-mat.stat-mech], 2018. (A078812)
  14. G. Failla, C. Peterson, R. Utano, Algorithms and basic asymptotics for generalized numerical semigroups in N^d, Semigroup Forum, Jan 31 2015, doi:10.1007/s00233-015-9690-8
  15. Sergio Falcon, On the k-Lucas numbers, Int. J. Cont. Math. Sci. 6 (2011) 1039-1050
  16. 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
  17. S. Falcon, Generalized Fibonacci Sequences Generated from a $ k $--Fibonacci Sequence, Journal of Mathematics Research Vol. 4, No. 2; April 2012;
  18. 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.
  19. S. Falcon, Relationships between Some k-Fibonacci Sequences, Applied Mathematics, 2014, 5, 2226-2234; doi:10.4236/am.2014.515216
  20. 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;
  21. 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);
  22. S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
  23. 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.
  24. S. Falcon, Generalized (k, r)–Fibonacci Numbers, Gen. Math. Notes, Vol. 25, No. 2, December 2014, pp.148-158;
  25. Sergio Falcon, The k–Fibonacci difference sequences, Chaos, Solitons & Fractals, Volume 87, June 2016, Pages 153–157.
  26. Sergio Falcon, On the complex k-Fibonacci numbers, Cogent Mathematics, (2016), 3: 1201944; doi:10.1080/23311835.2016.1201944
  27. Sergio Falcon, <a href="">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.
  28. 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)
  29. 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.
  30. 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.
  31. 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.
  32. Sergio Falcon, Angel Plaza, k-Fibonacci sequences modulo m, Chaos, Solitons & Fractals, Volume 41, Issue 1, 15 July 2009, Pages 497-504.
  33. Victor Falgas-Ravry and Emil R. Vaughan, On applications of Razborov's flag algebra calculus to extremal 3-graph theory, Arxiv preprint arXiv:1110.1623, 2011
  34. 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)
  35. Falgas-Ravry, Victor; Vaughan, Emil R. Applications of the semi-definite method to the Turán density problem for 3-graphs. Combin. Probab. Comput. 22 (2013), no. 1, 21-54. doi:10.1017/S0963548312000508
  36. Wenjie Fang, A partial order on Motzkin paths, Discrete Math., 343 (2020), #111802.
  37. Xin Fang and Ghislain Fourier, Torus fixed points in Schubert varieties and Genocchi numbers, arXiv:1504.03980.
  38. Reza Farhadian, A New Conjecture On the primes, Preprint, 2016;
  39. Bakir Farhi, A Study of a Curious Arithmetic Function Journal of Integer Sequences, Vol. 15 (2012), #12.3.1.
  40. 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)
  41. 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.
  42. G. Farkas, G. Kallos and G. Kiss, Large primes in generalized Pascal triangles Acta Univ. Sapientiae, Informatica, 3, 2 (2011) 158-171;
  43. Elin Farnell, Puzzle Pedagogy: A Use of Riddles in Mathematics Education, PRIMUS, July 2016, pp. 202-211; doi:10.1080/10511970.2016.1195465
  44. M. Farrokhi D. G., "Some Remarks On the Equation Fn = kFm In Fibonacci Numbers", J. Integer Sequences, Volume 10, 2007, Article 07.5.7.
  45. Joseph A. Farrow, A Monte Carlo Approach to the 4D Scattering Equations, arXiv:1806.02732 [hep-th], 2018. (A008292)
  46. 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"]
  47. James A. Farrugia, Brun's 1920 Theorem on Goldbach's Conjecture, Masters Thesis, Utah State University, All Graduate Theses and Dissertations (2018). 7153 HTML (A002375)
  48. Massimiliano Fasi, Gian Maria Negri Porzio, Determinants of Normalized Bohemian Upper Hessemberg Matrices, University of Manchester (England, 2019). Abstract (A000045, A000051, A001588, A001611, A003433, A052542)
  49. Nazim Fatès, Remarks on the Cellular Automaton Global Synchronisation Problem, in Cellular Automata and Discrete Complex Systems, Lecture Notes in Computer Science Volume 9099, 2015, pp 113-126.
  50. Nazim Fatès, Remarks on the cellular automaton global synchronisation problem–deterministic vs. stochastic models, Preprint 2017;
  51. Nazim Fatès, Biswanath Sethi, Sukanta Das, On the reversibility of ECAs with fully asynchronous updating: the recurrence point of view, Preprint, 2017, also doi:10.1007/978-3-319-73216-9_15 (A001612, A005251, A259967)
  52. F. Fauvet, L. Foissy, D. Manchon, The Hopf algebra of finite topologies and mould composition, arXiv preprint arXiv:1503.03820, 2015
  53. Frederic Fauvet, L Foissy, D Manchon, Operads of finite posets, arXiv preprint arXiv:1604.08149, 2016
  54. Bernadette Faye, Florian Luca, Pieter Moree, On the discriminator of Lucas sequences, arXiv:1708.03563 [math.NT], 2017.
  55. YN Fedorov, ANW Hone, Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties, arXiv preprint arXiv:1512.00056, 2015
  56. J. M. Fedou, Enumeration of skew Ferrers diagrams and basic Bessel functions, Journal of Statistical Planning and Inference, Volume 34, Issue 1, January 1993, Pages 107-123.
  57. J.-M. Fedou, G. Fici, Some remarks on differentiable sequences and recursivity, JIS 13 (2010) #10.3.2
  58. Greg Fee, Simon Plouffe, An efficient algorithm for the computation of Bernoulli numbers (2007), arXiv:math/0702300
  59. Ad Feelders, Linda C. van der Gaag, Learning Bayesian network parameters under order constraints, International Journal of Approximate Reasoning, Volume 42, Issues 1-2, May 2006, Pages 37-53.
  60. Uriel Feige, Tighter bounds for online bipartite matching, 2018. PDF (A000166, A000255, A180191)
  61. J. Feigenbaum, A. D. Jaggard, M. Schapira, Approximate Privacy: Foundations and Quantification, ACM Transactions on Algorithms (TALG), Volume 10 Issue 3, June 2014. Article No. 11
  62. Feigin, Evgeny, Degenerate flag varieties and the median Genocchi numbers. Math. Res. Lett. 18 (2011), no. 6, 1163-1178.
  63. E. Feigin, The median Genocchi numbers, Q-analogues and continued fractions, Arxiv preprint arXiv:1111.0740, 2011 and doi:10.1016/j.ejc.2012.06.004 Eur. J. Combinat. 33 (8) (2012) 1913-1918
  64. Pedro Feijao, Reconstruction of ancestral gene orders using intermediate genomes, BMC Bioinformatics 2015, 16 (Suppl 14):S3; doi:10.1186/1471-2105-16-S14-S3; Proceedings of the 13th Annual Research in Computational Molecular Biology (RECOMB) Satellite Workshop on Comparative Genomics: Bioinformatics
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  67. T. Feil, K. Hutson, R. M. Kretchmar, Tree Traversals and permutations, Congr. Numer. 172 (2005) 201-221
  68. G. Feinberg, K.-H. Lee, Homogeneous representations of KLR-algebras and fully commutative elements, arXiv preprint arXiv:1401.0845, 2014
  69. Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages; doi:10.1155/2011/539030.
  70. Andrew Feist, Fun with the (n) function.
  71. Leonid G. Fel, On Summatory Totient Functions (2008); arXiv:0802.0619
  72. D. P. Feldman and J. P. Crutchfield, Synchronizing to Periodicity: The Transient Information and Synchronization Time of Periodic Sequences, Santa Fe Institute Working Paper 02-08-043. arXiv:nlin/0208040. 2002.
  73. Stefan Felsner, Éric Fusy, Marc Noy et al., Bijections for Baxter Families and Related Objects (2008); arXiv:0803.1546, J. Comb. Theory A 118 (3) (2011) 993-1020 doi:10.1016/j.jcta.2010.03.017
  74. Felsner, Stefan; Trotter, William T., Posets and planar graphs. J. Graph Theory 49 (2005), no. 4, 273-284.
  75. S. Felsner, F. Zickfeld, On the number of planar orientations with prescribed degrees, Electr. J. Comb. 15 (1) (2008) R77, HTML
  76. Guglielmo Feltrin. Positive subharmonic solutions to superlinear ODEs with indefinite weight. arXiv:1701.06145 [math.CA], 2017.
  77. D.-J. Feng, P. Liardet and A. Thomas, Partition functions in numeration systems with bounded multiplicity, Uniform Distribution Theory, submitted 2013,
  78. Jishe Feng, The Hessenberg matrices and Catalan and its generalized numbers, arXiv preprint arXiv:1810.09170, 2018
  79. Jo Fenstad, Deciphering the hidden Pascal code of Guggenheim's quasichemical model. Overcoming computational limitations by use of Pascal's triangle, Journal of Mathematical Chemistry, Volume 42, Number 4 / November, 2007.
  80. V. Féray, Cyclic inclusion-exclusion, arXiv preprint arXiv:1410.1772, 2014.
  81. Svjetlan Feretić, The perimeter generating function for nondirected diagonally convex polyominoes, arXiv:1907.09409 [math.CO], 2019. (A269228)
  82. R. Feria-Puron, H. Perez-Roses, J. Ryan, Searching for Large Circulant Graphs, arXiv preprint arXiv:1503.07357, 2015
  83. R. Feria-Purón, J. Ryan, H. Pérez-Rosés, Searching for Large Multi-Loop Networks, Electronic Notes in Discrete Mathematics, Volume 46, September 2014, Pages 233-240.
  84. Mauricio Fernández, On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials, Journal of Elasticity (2019). doi:10.1007/s10659-019-09754-8 (A005043)
  85. Tim Fernando, Carl Vogel, Prior Probabilities of Allen Interval Relations over Finite Orders, 11th International Conference on Agents and Artificial Intelligence (NLPinAI 2019) Prague. PDF (A055203)
  86. Emmanuel Ferrand, "Deformations of the Taylor Formula", J. Integer Sequences, Volume 10, 2007, Article 07.1.7.
  87. Luca Ferrari, Some combinatorics related to central binomial coefficients: Grand-Dyck paths, coloured noncrossing partitions and signed pattern avoiding permutations (2008); arXiv:0806.0973 and Graphs. Comb. 26 (1) (2010) 51-70 doi:10.1007/s00373-010-0895-z
  88. Luca Ferrari, Schroder partitions, Schroder tableaux and weak poset patterns, arXiv preprint arXiv:1606.06624, 2016.
  89. Luca Ferrari, Enhancing the connections between patterns in permutations and forbidden configurations in restricted elections, arXiv:1906.10553 [math.CO], 2019. (A006012)
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  92. Luca Ferrari and Emanuele Munarini, Enumeration of edges in some lattices of paths, Arxiv preprint arXiv:1203.6792, 2012 and J. Int. Seq. 17 (2014) #14.1.5
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  98. José William Porras Ferreira, An Approach to Solve Erdös-Straus Conjecture, ResearchGate, 2017.
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  102. 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.
  103. J. Ferte, V. Pilaud and M. Pocchiola, On the number of simple arrangements of five double pseudolines. Discrete Comput. Geom. 45 (2011), 279-302.
  104. Guillaume Fertin, Samuele Giraudo, Sylvie Hamel, Stéphane Vialette, Unshuffling Permutations: Trivial Bijections and Compositions, International Conference on Theory and Applications of Models of Computation: Theory and Applications of Models of Computation (TAMC 2019). Springer, Cham. 242-261. doi:10.1007/978-3-030-14812-6_15
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  116. Peter E. Finch, From spin to anyon notation: The XXZ Heisenberg model as a D_3 (or su(2)_4) anyon chain, Arxiv preprint arXiv:1201.4470, 2012
  117. S. R. Finch, Mathematical Constants, Cambridge University Press (to appear), 2003. (Sample Essays and Supplementary Materials)
  118. Steven Finch, Pattern-Avoiding Permutations [There is a cached copy attached to A005802]
  119. S. R. Finch, Idempotents and Nilpotents Modulo n, arXiv:math.NT/0605019
  120. Steven Finch, Powers of Euler's q-Series, arXiv:math.NT/0701251.
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  122. S. Finch, <a href="">Toothpicks and live cells</a>
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  127. S. Finch, P. Sebah and Z.-Q. Bai, Odd Entries in Pascal's Trinomial Triangle arXiv:0802.2654
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  131. Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, PDF, 2006.
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  133. Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Masters Thesis, University of Waterloo, Ontario, Canada, 2018. PDF (A014486, A031443, A044951, A061854, A072600, A072601, A072602)
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  135. Lukas Finschi, A Graph Theoretical Approach for Reconstruction and Generation of Oriented Matroids, A dissertation submitted to the Swiss Federal Institute of Technology, Zurich for the degree of Doctor of Mathematics, 2001.
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  142. Georg Fischer, Richard J. Mathar, p-finite recurrences from sqrt generating functions, (2020). PDF (A026375, A078634, A122877, A242586)
  143. Johannes Fischer and Florian Kurpicz, Fast and Simple Parallel Wavelet Tree and Matrix Construction, arXiv:1702.07578 [cs.DS], 2017.
  144. Mareike Fischer, Extremal values of the Sackin balance index for rooted binary trees, arXiv:1801.10418 [q-bio.PE], 2018. (A000096, A003314, A299037)
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