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 which specific finite subsequences of the Collatz 3n + 1 function consists of n+2 terms. This has been verified for 0 < n < 51. For details see Links. - _Mike Winkler_, Oct 03 2014
%C A000045 a(4)=3 and a(6)=8 are the only Fibonacci numbers that are of the form prime+1. - _Emmanuel Vantieghem_, Oct 02 2014
%C A000045 a(1)=1=a(2), a(3)=2 are the only Fibonacci numbers that are of the form prime-1. - _Emmanuel Vantieghem_, Jun 07 2015
%C A000045 Any consecutive pair (m, k) of the Fibonacci sequence a(n) illustrates a fair equivalence between m miles and k kilometers. For instance, 8 miles ~ 13 km; 13 miles ~ 21 km. -_Lekraj Beedassy_, Oct 06 2014
%C A000045 Lim_{n -> oo} (log F(n+1)/log F(n))^n = e. - _Thomas Ordowski_, Oct 06 2014
%C A000045 a(n+1) counts closed walks on K_2, containing one loop on the other vertex. Equivalently the (1,1)_entry of A^(n+1) where the adjacency matrix of digraph is A=(0,1; 1,1). - _David Neil McGrath_, Oct 29 2014
%C A000045 a(n-1) counts closed walks on the graph G(1-vertex;l-loop,2-loop). - _David Neil McGrath_, Nov 26 2014
%C A000045 From _Tom Copeland_, Nov 02 2014: (Start)
%C A000045 Let P(x) = x/(1+x) with comp. inverse Pinv(x) = x/(1-x) = -P[-x], and C(x)= [1-sqrt(1-4x)]/2, an o.g.f. for the shifted Catalan numbers A000108, with inverse Cinv(x) = x * (1-x).
%C A000045 Fin(x) = P[C(x)] = C(x)/[1 + C(x)] is an o.g.f. for the Fine numbers, A000957 with inverse Fin^(-1)(x) = Cinv[Pinv(x)] = Cinv[-P(-x)].
%C A000045 Mot(x) = C[P(x)] = C[-Pinv(-x)] gives an o.g.f. for shifted A005043, the Motzkin or Riordan numbers with comp. inverse Mot^(-1)(x) = Pinv[Cinv(x)] = (x - x^2) / (1 - x + x^2) (cf. A057078).
%C A000045 BTC(x) = C[Pinv(x)] gives A007317, a binomial transform of the Catalan numbers, with BTC^(-1)(x) = P[Cinv(x)].
%C A000045 Fib(x) = -Fin[Cinv(Cinv(-x))] = -P[Cinv(-x)] = x + 2 x^2 + 3 x^3 + 5 x^4 + ... = (x+x^2)/[1-x-x^2] is an o.g.f. for the shifted Fibonacci sequence A000045, so the comp. inverse is Fib^(-1)(x) = -C[Pinv(-x)] = -BTC(-x) and Fib(x) = -BTC^(-1)(-x).
%C A000045 Generalizing to P(x,t) = x /(1 + t*x) and Pinv(x,t) = x /(1 - t*x) = -P(-x,t) gives other relations to lattice paths, such as the o.g.f. for A091867, C[P[x,1-t]], and that for A104597, Pinv[Cinv(x),t+1].
%C A000045 (End)
%C A000045 In keeping with historical accounts (see the references by P. Singh and S. Kak), the generalized Fibonacci sequence a, b, a + b, a + 2b, 2a + 3b, 3a + 5b, ... can also be described as the Gopala-Hemachandra numbers H(n) = H(n-1) + H(n-2), with F(n) = H(n) for a = b = 1, and Lucas sequence L(n) = H(n) for a = 2, b = 1. - _Lekraj Beedassy_, Jan 11 2015
%C A000045 D. E. Knuth writes: "Before Fibonacci wrote his work, the sequence F_{n} had already been discussed by Indian scholars, who had long been interested in rhythmic patterns that are formed from one-beat and two-beat notes. The number of such rhythms having n beats altogether is F_{n+1}; therefore both Gopāla (before 1135) and Hemachandra (c. 1150) mentioned the numbers 1, 2, 3, 5, 8, 13, 21, ... explicitly." (TAOCP Vol. 1, 2nd ed.) - _Peter Luschny_, Jan 11 2015
%C A000045 F(n+1) equals the number of binary words of length n avoiding runs of zeros of odd lengths. - _Milan Janjic_, Jan 28 2015
%C A000045 From _Russell Jay Hendel_, Apr 12 2015: (Start)
%C A000045 We prove Conjecture 1 of Rashid listed in the Formula section.
%C A000045 We use the following notation: F(n)=A000045(n), the Fibonacci numbers, and L(n) = A000032(n), the Lucas numbers. The fundamental Fibonacci-Lucas recursion asserts that G(n) = G(n-1)+ G(n-2), with "L" or "F" replacing "G".
%C A000045 We need the following prerequisites which we label (A), (B),(C), (D). The prerequisites are formulas in the Koshy book listed in the References section. (A) F(m-1)+F(m+1) = L(m) (Koshy, p. 97, #32), (B) L(2m)+2(-1)^m = L(m)^2 (Koshy p. 97, #41), (C) F(m+k)F(m-k) = (-1)^n F(k)^2 (Koshy, p. 113, #24, Tagiuri's identity), and (D) F(n)^2+F(n+1)^2 = F(2n+1) (Koshy, p. 97, #30).
%C A000045 We must also prove (E), L(n+2) F(n-1) = F(2n+1)+2(-1)^n. To prove (E), first note that by (A), proof of (E) is equivalent to proving that F(n+1)F(n-1) + F(n+3)F(n-1) = F(2n+1)+2(-1)^n. But by (C) with k=1, we have F(n+1)F(n-1) = F(n)^2 +(-1)^n. Applying (C) again with k=2 and m=n+1, we have F(n+3)F(n-1) = F(n+1)+(-1)^n. Adding these two applications of (C) together and using (D) we have, F(n+1)F(n-1) + F(n+3)F(n-1) = F(n)^2 + F(n+1)^2 + 2(-1)^n = F(2n+1)+2(-1)^n, completing the proof of (E).
%C A000045 We now prove Conjecture 1. By (A) and the Fibonacci-Lucas recursion, we have F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4) = [F(2n+1)+F(2n+3)] + [F(2n+2)+F(2n+4)] = L(2n+2)+L(2n+3)=L(2n+4). But then by (B), with m=2n+4, we have sqrt(L(2n+4)+2(-1)^n)) = L(n+2). Finally by (E), we have L(n+2) F(n-1)= F(2n+1)+2*(-1)^n. Dividing both sides by F(n-1), we have (F(2n+1)+2*(-1)^n)/F(n-1) = L(n+2) = sqrt(F(2n+1)+F(2n+2)+F(2n+3)+F(2n+4)+2(-1)^n), as required.
%C A000045 (End)
%C A000045 In Fibonacci's Liber Abaci the rabbit problem appears in the translation of L. E. Sigler on pp. 404-405, and a remark [27] on p. 637. - _Wolfdieter Lang_, Apr 17 2015
%C A000045 a(n) counts partially ordered partitions of (n-1) into parts 1,2,3 where only the order of adjacent 1's and 2's are unimportant. (See example.) - _David Neil McGrath_, Jul 27 2015
%C A000045 F(n) divides F(nk). Proved by Marjorie Bicknell and Verner E Hoggatt Jr. - _Juhani Heino_, Aug 24 2015
%C A000045 F(n) is the number of UDU-equivalence classes of ballot paths of length n. Two ballot paths of length n with steps U = (1,1), D = (1,-1) are UDU-equivalent whenever the positions of UDU are the same in both paths. - _Kostas Manes_, Aug 25 2015
%C A000045 Cassini's identity F(2n+1) * F(2n+3) = F(2n+2)^2 + 1 is the basis for a geometrical paradox (or dissection fallacy) in A262342. - _Jonathan Sondow_, Oct 23 2015
%C A000045 For n >= 4, F(n) is the number of up-down words on alphabet {1,2,3} of length n-2. - _Ran Pan_, Nov 23 2015
%C A000045 F(n+2) is the number of terms in p(n), where p(n)/q(n) is the n-th convergent of the formal infinite continued fraction [a(0),a(1),...]; e.g., p(3) = a(0)a(1)a(2)a(3) + a(0)a(1) + a(0)a(3) + a(2)a(3) + 1 has F(5) terms. Also, F(n+1) is the number of terms in q(n). - _Clark Kimberling_, Dec 23 2015
%C A000045 F(n+1) (for n >= 1) is the permanent of an n X n matrix M with M(i,j)=1 if |i-j| <= 1 and 0 otherwise. - _Dmitry Efimov_, Jan 08 2016
%C A000045 A trapezoid has three sides of lengths in order F(n), F(n+2), F(n). For increasing n a very close approximation to the maximum area will have the fourth side equal to 2*F(n+1). For a trapezoid with lengths of sides in order F(n+2), F(n), F(n+2), the fourth side will be F(n+3). - _J. M. Bergot_, Mar 17 2016
%C A000045 (1) Join two triangles with lengths of sides L(n), F(n+3), L(n+2) and F(n+2), L(n+1), L(n+2) (where L(n)=A000032(n)) along the common side of length L(n+2) to create an irregular quadrilateral. Its area is approximately (5*F(2*n-1) - (F(2*n-7) - F(2*n-13))/5. (2) Join two triangles with lengths of sides L(n), F(n+2), F(n+3) and L(n+1), F(n+1, F(n+3) along the common side F(n+3) to form an irregular quadrilateral. Its area is approximately 4*F(2*n-1) - 2*(F(2*n-7) + F(2*n-18)). - _J. M. Bergot_, Apr 06 2016
%C A000045 From _Clark Kimberling_, Jun 13 2016: (Start)
%C A000045 Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*.
%C A000045 Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2, x}, g(3) = {3, 2x, x+1, x^2}, etc.
%C A000045 Let T(r) be the tree obtained by substituting r for x.
%C A000045 If a positive integer N is not a square and r = sqrt(N), then the number of (not necessarily distinct) integers in g(n) is A000045(n), for n > = 1. See A274142. (End)
%C A000045 Consider the partitions of n, with all summands initially listed in nonincreasing order. Freeze all the 1's in place and then allow all the other summands to change their order, without displacing any of the 1's. The resulting number of arrangements is a(n+1). - _Gregory L. Simay_, Jun 14 2016
%C A000045 Limit of the matrix power M^k shown in A163733, Sep 14 2016; as k-->inf. results in a single column vector equal to the Fibonacci sequence. - _Gary W. Adamson_, Sep 19 2016
%C A000045 F(n) and Lucas numbers L(n), being related by the formulas F(n) = (F(n-1) + L(n-1))/2 and L(n) = 2 F(n+1) - F(n), are a typical pair of "autosequences" (see the link to OEIS Wiki). - _Jean-François Alcover_, Jun 10 2017
%C A000045 Also the number of independent vertex sets and vertex covers in the (n-2)-path graph. - _Eric W. Weisstein_, Sep 22 2017
%C A000045 Shifted numbers of {UD, DU, FD, DF}-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are P-equivalent iff the positions of pattern P are identical in these paths. - _Sergey Kirgizov_, Apr 08 2018
%C A000045 For n>0, F(n) = the number of Markov equivalence classes with skeleton the path on n nodes. See Theorem 2.1 in the article by A. Radhakrishnan et al. below. - _Liam Solus_, Aug 23 2018
%C A000045 For n >= 2, also: number of terms in A032858 (every other base-3 digit is strictly smaller than its neighbors) with n-2 digits in base 3. - _M. F. Hasler_, Oct 05 2018
%C A000045 F(n+1) is the number of fixed points of the Foata transformation on S_n. - _Kevin Long_, Oct 17 2018
%C A000045 F(n+2) is the dimension of the Hecke algebra of type A_n with independent parameters (0,1,0,1,...) or (1,0,1,0,...). See Corollary 1.5 in the link "Hecke algebras with independent parameters". - _Jia Huang_, Jan 20 2019
%D A000045 Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 78-79. Zentralblatt MATH, Zbl 1097.11516.
%D A000045 Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17.
%D A000045 Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7,  No. 38, 2012, pp. 1871-1876.
%D A000045 Mohammad K. Azarian, Fibonacci Identities as Binomial Sums II, International Journal of Contemporary Mathematical Sciences, Vol. 7,  No. 42, 2012, pp. 2053-2059.
%D A000045 Mohammad K. Azarian, Identities Involving Lucas or Fibonacci and Lucas Numbers as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 45, 2012, pp. 2221-2227.
%D A000045 P. Bachmann, Niedere Zahlentheorie (1902, 1910), reprinted Chelsea, NY, 1968, vol. 2, p. 70.
%D A000045 R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 84.
%D A000045 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 4.
%D A000045 Marjorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
%D A000045 Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 24 (Ex. 18), 489, 541.
%D A000045 A. Cayley, Theorems in Trigonometry and on Partitions, Messenger of Mathematics, 5 (1876), pp. 164, 188 = Mathematical Papers Vol. 10, n. 634, p. 16.
%D A000045 D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulas of Scaled Fibonacci Numbers (submitted 2012).
%D A000045 B. A. Davey and H. A. Priestley, Introduction to Lattices and Order (2nd edition), CUP, 2002. (See Exercise 1.15.)
%D A000045 B. Davis, 'The law of first digits' in 'Science Today' (subsequently renamed '2001') March 1980 p. 55, Times of India, Mumbai.
%D A000045 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.2.
%D A000045 R. P. Grimaldi, Compositions without the summand 1, Proceedings Thirty-second Southeastern International Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, LA, 2001). Congr. Numer. 152 (2001), 33-43.
%D A000045 H. Halberstam and K. F. Roth, Sequences, Oxford, 1966; see Appendix.
%D A000045 S. Happersett, "Mathematical meditations", Journal of Mathematics and the Arts, 1 (2007), 29 - 33.
%D A000045 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954; see esp. p. 148.
%D A000045 J. Hermes, Anzahl der Zerlegungen einer ganzen rationalen Zahl in Summanden, Math. Ann., 45 (1894), 371-380.
%D A000045 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers. Houghton, Boston, MA, 1969.
%D A000045 E. Horowitz and S. Sahni, Fundamentals of Data Structures, Computer Science Press, 1976; p. 338.
%D A000045 P. W. Kasteleyn, The statistics of dimers on a lattice. I. The number of dimer arrangements on a quadratic lattice, Physica, 27 (1961), 1209-1225.
%D A000045 M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 63.
%D A000045 C. Kicey and K. Klimko, Some geometry of Pascal's triangle, Pi Mu Epsilon Journal, 13(4):229-245 (2011).
%D A000045 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 78; Vol. 3, Section 6.2.1.
%D A000045 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", John Wiley and Sons, 2001.
%D A000045 Leonardo of Pisa [Leonardo Pisano], Liber Abaci [The Book of Calculation], 1202.
%D A000045 D. Litchfield, D. Goldenheim and C. H. Dietrich, Euclid, Fibonacci and Sketchpad, Math. Teacher, 90 (1997).
%D A000045 Lukovits et al., Nanotubes: Number of Kekulé structures and aromaticity, J. Chem. Inf. Comput. Sci, (2003), vol. 43, 609-614. See eq. 2 on page 610.
%D A000045 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44, 410-414 (2004). See Table 1, second column.
%D A000045 B. Malesevic: Some combinatorial aspects of differential operation composition on the space R^n, Univ. Beograd, Publ. Elektrotehn. Fak., Ser. Mat. 9 (1998), 29-33.
%D A000045 G. Mantel, Resten van wederkeerige Reeksen, Nieuw Archief v. Wiskunde, 2nd series, I (1894), 172-184.
%D A000045 C. N. Menhinick, The Fibonacci Resonance and other new Golden Ratio discoveries, Onperson, (2015), pages 200-206.
%D A000045 A. Milicevic and N. Trinajstic, "Combinatorial Enumeration in Chemistry", Chem. Modell., Vol. 4, (2006), pp. 405-469.
%D A000045 S. Mneimneh, Fibonacci in The Curriculum: Not Just a Bad Recurrence, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, Pages 253-258.
%D A000045 Hilary I. Okagbue, Muminu O. Adamu, Sheila A. Bishop, Abiodun A. Opanuga, Digit and Iterative Digit Sum of Fibonacci numbers, their identities and powers, International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 6 (2016) pp 4623-4627.
%D A000045 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 49.
%D A000045 Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics, Sterling Publ., NY, 2009, page 274.
%D A000045 A. S. Posamentier & I. Lehmann, The Fabulous Fibonacci Numbers, Prometheus Books, Amherst, NY 2007.
%D A000045 P. Ribenboim, The New Book of Prime Number Records, Springer, 1996.
%D A000045 J. Riordan, An Introduction to Combinatorial Analysis, Princeton University Press, Princeton, NJ, 1978.
%D A000045 A. M. Robert, A Course in p-adic Analysis, Springer-Verlag, 2000; p. 213.
%D A000045 J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 288.
%D A000045 Manfred R. Schroeder, "Number Theory in Science and Communication", 5th ed., Springer-Verlag, 2009
%D A000045 L. E. Sigler, Fibonacci's Liber Abaci, Springer, 2003, pp. 404-405 and [26] on p. 627.
%D A000045 Simson, [No first name given], An explanation of an obscure passage in Albert Girard's Commentary ..., Phil. Trans. Royal Soc., 10 (1753), 430-433.
%D A000045 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A000045 S. Vajda, Fibonacci and Lucas numbers and the Golden Section, Ellis Horwood Ltd., Chichester, 1989.
%D A000045 Van Ravenstein, Tony. "The three gap theorem (Steinhaus conjecture)." Journal of the Australian Mathematical Society (Series A) 45.03 (1988): 360-370.
%D A000045 N. N. Vorob'ev, Chisla fibonachchi [Russian], Moscow, 1951. English translation, Fibonacci Numbers, Blaisdell, New York and London, 1961.
%D A000045 N. N. Vorobiev, Fibonacci Numbers, Birkhauser (Basel;Boston) 2002.
%D A000045 L. C. Washington, Benford's Law for Fibonacci and Lucas numbers, Fib. Quaterly, 1981.
%D A000045 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 61-7, Penguin Books 1987.
%D A000045 R. Witula, D. Slota, delta-Fibonacci Numbers, Appl. Anal. Discrete Math., 3 (2009), 310-329.
%H A000045 N. J. A. Sloane, <a href="/A000045/b000045.txt">The first 2000 Fibonacci numbers: Table of n, F(n) for n = 0..2000</a>
%H A000045 Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="http://dx.doi.org/10.1016/j.disc.2014.06.026">Colored compositions, Invert operator and elegant compositions with the "black tie"</a>, Discrete Math. 335 (2014), 1--7. MR3248794.
%H A000045 Matt Anderson, Jeffrey Frazier and Kris Popendorf, <a href="http://library.thinkquest.org/27890/theSeries.html">The Fibonacci series: the section index</a> [broken link]
%H A000045 P. G. Anderson, <a href="http://www.cs.rit.edu/~pga/Fibo/fact_sheet.html">Fibonacci Facts</a>
%H A000045 George E. Andrews, <a href="https://www.fq.math.ca/Papers1/42-1/quartgranville01_2004.pdf">Fibonacci numbers and the Rogers-Ramanujan identities</a>, Fibonacci Quart. 42 (2004), no. 1, 3-19. MR2060558(2005b:11161).
%H A000045 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>
%H A000045 Shaun V. Ault and Charles Kicey, <a href="http://dx.doi.org/10.1016/j.disc.2014.05.020">Counting paths in corridors using circular Pascal arrays</a>, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.
%H A000045 Ovidiu Bagdasar, Eve Hedderwick, Ioan-Lucian Popa, <a href="https://doi.org/10.1016/j.endm.2018.05.011">On the ratios and geometric boundaries of complex Horadam sequences</a>, Electronic Notes in Discrete Mathematics (2018) Vol. 67, 63-70.
%H A000045 S. Barbero, U. Cerruti, N. Murru, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barbero/barbero5.html">Transforming Recurrent Sequences by Using the Binomial and Invert Operators</a>, J. Int. Seq. 13 (2010) # 10.7.7, example 17.
%H A000045 J.-L. Baril, <a href="http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf">Classical sequences revisited with permutations avoiding dotted pattern</a>, Electronic Journal of Combinatorics, 18 (2011), #P178.
%H A000045 Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian, <a href="https://arxiv.org/abs/1804.01293">Enumeration of Łukasiewicz paths modulo some patterns</a>, arXiv:1804.01293 [math.CO], 2018.
%H A000045 Marilena Barnabei, Flavio Bonetti, and Niccolò Castronuovo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barnabei/barnabei5.html">Motzkin and Catalan Tunnel Polynomials</a>, J. Int. Seq., Vol. 21 (2018), Article 18.8.8.
%H A000045 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Barry/barry84.html">A Catalan Transform and Related Transformations on Integer Sequences</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%H A000045 Paul Barry, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Barry/barry91.html">On Integer-Sequence-Based Constructions of Generalized Pascal Triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
%H A000045 P. Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.
%H A000045 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A000045 A. T. Benjamin, A. K. Eustis, M. A. Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS