

A001906


F(2n) = bisection of Fibonacci sequence: a(n) = 3*a(n1)  a(n2).
(Formerly M2741 N1101)


402



0, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, 46368, 121393, 317811, 832040, 2178309, 5702887, 14930352, 39088169, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717, 591286729879, 1548008755920
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OFFSET

0,3


COMMENTS

Apart from initial term, same as A088305.
Second column of array A102310 and of A028412.
Numbers n such that 5*n^2 + 4 is a square.  Gregory V. Richardson, Oct 13 2002
Apart from initial terms, also Pisot sequences E(3,8), P(3,8), T(3,8). See A008776 for definitions of Pisot sequences.
Binomial transform of A000045.  Paul Barry, Apr 11 2003
Number of walks of length 2n+1 in the path graph P_4 from one end to the other one. Example: a(2)=3 because in the path ABCD we have ABABCD, ABCBCD and ABCDCD.  Emeric Deutsch, Apr 02 2004
Simplest example of a secondorder recurrence with the sixth term a square.
Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 5 and s(i)  s(i1) = 1 for i = 1,2,...,2n, s(0) = 1, s(2n) = 3.  Lekraj Beedassy, Jun 11 2004
a(n) (for n > 0) is the smallest positive integer that cannot be created by summing at most n values chosen among the previous terms (with repeats allowed).  Andrew Weimholt, Jul 20 2004
All nonnegative integer solutions of Pell equation b(n)^2  5*a(n)^2 = +4 together with b(n) = A005248(n), n >= 0.  Wolfdieter Lang, Aug 31 2004
a(n+1) is a Chebyshev transform of 3^n (A000244), where the sequence with g.f. G(x) is sent to the sequence with g.f. (1/(1+x^2))G(x/(1+x^2)).  Paul Barry, Oct 25 2004
a(n) is the number of distinct products of matrices A, B, C, in (A+B+C)^n where commutator [A,B] = 0 but C does not commute with A or B.  Paul D. Hanna and Max Alekseyev, Feb 01 2006
Number of binary words with exactly k1 strictly increasing runs. Example: a(3)=F(6)=8 because we have 00,10,11,001,010,101,011 and 0101. Column sums of A119900.  Emeric Deutsch, Jul 23 2006
See Table 1 on page 411 of Lukovits and Janezic paper.  Parthasarathy Nambi, Aug 22 2006
Inverse: With phi = (sqrt(5) + 1)/2, log_phi((sqrt(5) a(n) + sqrt(5 a(n)^2 + 4))/2) = n.  David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007
[1,3,8,21,55,144,...] is the Hankel transform of [1,1,4,17,75,339,1558,...](see A026378).  Philippe Deléham, Apr 13 2007
The Diophantine equation a(n) = m has a solution (for m >= 1) if and only if floor(arcsinh(sqrt(5)*m/2)/log(phi)) <> floor(arccosh(sqrt(5)*m/2)/log(phi)) where phi is the golden ratio. An equivalent condition is A130259(m) = A130260(m).  Hieronymus Fischer, May 25 2007
a(n+1) = AB^(n)(1), n >= 0, with compositions of Wythoff's complementary A(n):=A000201(n) and B(n)=A001950(n) sequences. See the W. Lang link under A135817 for the Wythoff representation of numbers (with A as 1 and B as 0 and the argument 1 omitted). E.g., 1=`1`, 3=`10`, 8=`100`, 21=`1000`,..., in Wythoff code.
Equals row sums of triangles A140069, A140736 and A140737.  Gary W. Adamson, May 25 2008
a(n) is also the number of idempotent orderpreserving partial transformations (of an nelement chain) of width n (width(alpha) = max(Im(alpha))). Equivalently, it is the number of idempotent orderpreserving full transformations (of an nelement chain).  Abdullahi Umar, Sep 08 2008
a(n) is the number of ways that a string of 0,1 and 2 of size (n1) can be arranged with no 12pairs.  Udita Katugampola, Sep 24 2008
Starting with offset 1 = row sums of triangle A175011.  Gary W. Adamson, Apr 03 2010
As a fraction: 1/71 = 0.01408450... or 1/9701 = 0.0001030821....  Mark Dols, May 18 2010
Sum of the products of the elements in the compositions of n (example for n=3: the compositions are 1+1+1, 1+2, 2+1, and 3; a(3) = 1*1*1 + 1*2 + 2*1 + 3 = 8).  Dylon Hamilton, Jun 20 2010, Geoffrey Critzer, Joerg Arndt, Dec 06 2010
a(n) relates to regular polygons with even numbers of edges such that Product_{k=1..(n2)/2} (1 + 4*cos^2 k*Pi/n) = even indexed Fibonacci numbers with a(n) relating to the 2*ngons. The constants as products = roots to even indexed rows of triangle A152063. For example: a(5) = 55 satisfies the product formula relating to the 10gon.  Gary W. Adamson, Aug 15 2010
Alternatively, product of roots to x^4  12x^3 + 51x^2  90x + 55, (10th row of triangle A152063) = (4.618...)*(3.618...)*(2.381...)*(1.381...) = 55.  Gary W. Adamson, Aug 15 2010
a(n) is the number of generalized compositions of n when there are i different types of i, (i=1,2,...).  Milan Janjic, Aug 26 2010
Starting with "1" = row sums of triangle A180339, and eigensequence of triangle A137710.  Gary W. Adamson, Aug 28 2010
a(2) = 3 is the only prime.
Number of nonisomorphic graded posets with 0 and uniform hasse graph of rank n > 0, with exactly 2 elements of each rank level above 0. (Uniform used in the sense of Retakh, Serconek, and Wilson. Graded used in Stanley's sense that every maximal chain has the same length n.)  David Nacin, Feb 13 2012
Pisano period lengths: 1, 3, 4, 3, 10, 12, 8, 6, 12, 30, 5, 12, 14, 24, 20, 12, 18, 12, 9, 30, ...  R. J. Mathar, Aug 10 2012
Solutions (x, y) = (a(n), a(n+1)) satisfying x^2 + y^2 = 3xy + 1.  Michel Lagneau, Feb 01 2014
For n >= 1, a(n) equals the number of 01avoiding words of length n1 on alphabet {0,1,2}.  Milan Janjic, Jan 25 2015
With a(0) = 0, for n > 1, a(n) is the smallest number not already in the sequence such that a(n)^2  a(n1)^2 is a Fibonacci number.  Derek Orr, Jun 08 2015
Let T be the tree generated by these rules: 0 is in T, and if p is in T, then p + 1 is in T and x*p is in T and y*p is in T. The nth generation of T consists of A001906(n) polynomials, for n >= 0.  Clark Kimberling, Nov 24 2015
For n > 0, a(n) = exactly the maximum area of a quadrilateral with sides in order of lengths F(n), F(n), L(n), and L(n) with L(n)=A000032(n).  J. M. Bergot, Jan 20 2016
a(n) = twice the area of a triangle with vertices at (L(n+1), L(n+2)), (F(n+1), F(n+1)), and (L(n+2), L(n+1)), with L(n)=A000032(n).  J. M. Bergot, Apr 20 2016
Except for the initial 0, this is the pINVERT of (1,1,1,1,1,...) for p(S) = 1  S  S^2; see A291000.  Clark Kimberling, Aug 24 2017
a(n+1) is the number of spanning trees of the graph T_n, where T_n is a sequence of n triangles, where adjacent triangles share an edge.  Kevin Long, May 07 2018
a(n) is the number of ways to partition [n] such that each block is a run of consecutive numbers, and each block has a fixed point, e.g., for n=3, 123 with 1 and 3 as fixed points is valid, but 132 is not valid as 1 and 3 do not form a run. Consequently, a(n) also counts the spanning trees of the graph given by taking a path with n vertices and adding another vertex adjacent to all of them.  Kevin Long, May 11 2018
From Wolfdieter Lang, May 31 2018: (Start)
The preceding comment can be paraphrased as follows. a(n) is the row sum of the array A305309 for n >= 1. The array A305309(n, k) gives the sum of the products of the block lengths of the set partition of [n] := {1, 2, ..., n} with A048996(n, k) blocks of consecutive numbers, corresponding to the compositions obtained from the kth partition of n in AbramowitzStegun order. See the comments and examples at A305309.
{a(n)} also gives the infinite sequence of nonnegative numbers k for which k * k*phi < 1/sqrt(5), where the irrational number phi = A001622 (golden section), and x is the absolute value of the difference between x and the nearest integer. See, e.g., the Havil reference, pp. 171172. (End)
This Chebyshev sequence a(n) = S(n1, 3) (see a formula below) is related to the bisection of Fibonacci sequences {F(a,b;n)}_{n>=0} with input F(a,b;0) = a and F(a,b;1) = b, by F(a,b;2*k) = (a+b)*S(k1, 3)  a*S(k2, 3) and F(a,b;2*k+1) = b*S(k, 3) + (ab)*S(k1, 3), for k >= 0, and S(2, 3) = 1. Proof via the o.g.f.s GFeven(a,b,t) = (a  t*(2*ab))/(1  3*t + t^2) and GFodd(a,b,t) = (b + t*(ab))/(1  3*t + t^2). The special case a = 0, b = 1 gives back F(2*k) = S(k1, 3) = a(k).  Wolfdieter Lang, Jun 07 2019
a(n) is the number of tilings of two n X 1 rectangles joined orthogonally at a common endsquare (so to have 2n1 squares in a rightangle V shape) with only 1 X 1 and 2 X 1 tiles. This is a consequence of F(2n) = F(n+1)*F(n) + F(n)*F(n1).  Nathaniel Gregg, Oct 10 2021
These are the denominators of the upper convergents to the golden ratio, tau; they are also the numerators of the lower convergents (viz. 1/1 < 3/2 < 8/5 < 21/13 < ... < tau < ... 13/8 < 5/3 < 2/1).  Clark Kimberling, Jan 02 2022


REFERENCES

Mohammad K. Azarian, The Generating Function for the Fibonacci Sequence, Missouri Journal of Mathematical Sciences, Vol. 2, No. 2, Spring 1990, pp. 7879. Zentralblatt MATH, Zbl 1097.11516.
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 1217.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 2,5,6,14,33,55.
R. J. Douglas, Tournaments that admit exactly one Hamiltonian cycle, Proc. London Math. Soc., 21 (1970), 716730.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
A. Gerardin, Reply to Query 4389, L'Intermédiaire des Mathématiciens, 22 (1915), 23.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 171172.
Howie, J. M. Combinatorial and probabilistic results in transformation semigroups. Words, languages and combinatorics, II (Kyoto, 1992), 200206, World Sci. Publ., River Edge, NJ, (1994).
Laradji, A. and Umar, A. Combinatorial results for semigroups of orderpreserving full transformations. Semigroup Forum 72 (2006), 5162.
I. Lukovits, A. Graovac, E. Kalman, G. Kaptay, P. Nagy, S. Nikolic, J. Sytchev and N. Trinajstich, "Nanotubes: Number of Kekulé Structures and Aromaticity", J. Chem. Inf. Comput. Sci, vol. 43 (2003), pp. 609614. See Equation 6 on page 611.
T. Mansour, M. Shattuck, A statistic on ncolor compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 127140.
H. Mathieu, Query 3932, L'Intermédiaire des Mathématiciens, 18 (1911), 222.  N. J. A. Sloane, Mar 08 2022
I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 101.
Paulo Ribenboim, Primes in Lucas sequences (Chap 4), in 'My Numbers, My Friends', SpringerVerlag 2000 NY, page 27.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96100.


LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..2388 (terms 0..200 from T. D. Noe)
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Colored compositions, Invert operator and elegant compositions with the "black tie", Discrete Math. 335 (2014), 17. MR3248794.
Marco Abrate, Stefano Barbero, Umberto Cerruti, and Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.
K. Andersen, L. Carbone, and D. Penta, KacMoody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245278, 2011. See Section 9.
Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.
C. Banderier, M. BousquetMélou, A. Denise, P. Flajolet, D. Gardy and D. GouyouBeauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(13), March 2002, pp. 2955.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 5.
A. Bremner and N. Tzanakis, Lucas sequences whose 12th or 9th term is a square, arXiv:math/0405306 [math.NT], 2004.
David Broadhurst, Multiple Landen values and the tribonacci numbers, arXiv:1504.05303 [hepth], 2015.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89102.
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
Marc Chamberland and Christopher French, Generalized Catalan Numbers and Generalized Hankel Transformations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.1.
A. Collins et al., Binary words, ncolor compositions and bisection of the Fibonacci numbers, Fib. Quarterly, 51 (2013), 130136.
Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, Catalan Numbers, the Hankel Transform and Fibonacci Numbers, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.3
Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 15991607. See Cor. 3.7(e).
S. Falcon, Relationships between Some kFibonacci Sequences, Applied Mathematics, 2014, 5, 22262234.
Sergio Falcon, Catalan transform of the KFibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827832. doi:10.4134/CKMS.2013.28.4.827.
S. Falcon, Iterated Binomial Transforms of the kFibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
R. Flórez, R. A. Higuita, and A. Mukherjee, Alternating Sums in the Hosoya Polynomial Triangle, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).
Achille Frigeri, A note on Fibonacci number of even index, arXiv:1705.08305 [math.NT], 2017.
M. R. Garey, On enumerating tournaments that admit exactly one Hamiltonian circuit, J. Combin. Theory, B 13 (1972), 266269.
Dale Gerdemann, Fractal images from (3,1) recursion, YouTube Video, Oct 30 2014.
I. M. Gessel and Ji Li, Compositions and Fibonacci identities, J. Int. Seq. 16 (2013), 13.4.5.
A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding Part 1 Part 2, Fib. Quart., 9 (1971), 277295, 298.
Yh. Guo, Some nColor Compositions, J. Int. Seq. 15 (2012) 12.1.2, eq (2).
Edyta Hetmaniok, Bozena Piatek, and Roman Wituła, Binomials Transformation Formulae of Scaled Fibonacci Numbers, Open Math. 15 (2017), 477485.
A. F. Horadam, Special Properties of the Sequence W(n){a,b; p,q}, Fib. Quart., Vol. 5, No. 5 (1967), pp. 424434. Case a=0,b=1; p=3, q=1.
J. M. Howie, Products of idempotents in certain semigroups of transformations, Proc. Edinburgh Math. Soc. 17 (1971), 223236.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 147
Milan Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5.
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
J. Jina and P. Trojovsky, On determinants of some tridiagonal matrices connected with Fibonacci numbers, International Journal of Pure and Applied Mathematics 88:4 (2013), 569575.
Tanya Khovanova, Recursive Sequences
E. Kilic, Y. T. Ulutas, and N. Omur, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, Table 1, k=t=1.
Seong Ju Kim, R. Stees, and L. Taalman, Sequences of Spiral Knot Determinants, Journal of Integer Sequences, Vol. 19 (2016), #16.1.4.
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Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408419. Eq. (44) lhs, m=5.
IoanaClaudia Lazăr, Lucas sequences in tuniform simplicial complexes, arXiv:1904.06555 [math.GR], 2019.
I. Lukovits and D. Janezic, Enumeration of conjugated circuits in nanotubes, J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410414.
G. Narang and A. K. Agarwal, Lattice paths and ncolor compositions, Discr. Math., 308 (2008), 17321740.
László Németh and László Szalay, Sequences Involving Square ZigZag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
YunTak Oh, Hosho Katsura, HyunYong Lee, and Jung Hoon Han, Proposal of a spinone chain model with competing dimer and trimer interactions, arXiv:1709.01344 [condmat.strel], 2017.
J. Pan, Multiple Binomial Transforms and Families of Integer Sequences, J. Int. Seq. 13 (2010), 10.4.2. Absolute values of F^(2).
C. Pita, On sFibonomials, J. Int. Seq. 14 (2011) # 11.3.7.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 20102011.
John Riordan, Letter to N. J. A. Sloane, Nov. 1970
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their Nnumbers, not their Anumbers.
J. Salas and A. D. Sokal, Transfer Matrices and PartitionFunction Zeros for Antiferromagnetic Potts Models. V. Further Results for the SquareLattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279373, arXiv:0711.1738. Mentions this sequence.
Luigi Santocanale, On discrete idempotent paths, arXiv:1906.05590 [math.LO], 2019.
N. J. A. Sloane, Transforms
Michael Somos, In the Elliptic Realm.
Ryan Stees, Sequences of Spiral Knot Determinants, Senior Honors Projects, Paper 84, James Madison Univ., May 2016.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions.
Roman Witula, Binomials transformation formulae of scaled Lucas numbers, Demonstratio Math. 46 (2013), 1527.
R. Witula and Damian Slota, deltaFibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310329, MR2555042
Index entries for "core" sequences
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (3,1).
Index entries for twoway infinite sequences


FORMULA

G.f.: x / (1  3*x + x^2).  Simon Plouffe in his 1992 dissertation
a(n) = 3*a(n1)  a(n2) = A000045(2*n).
a(n) = a(n).
a(n) = A060921(n1, 0), n >= 1.
a(n) = sqrt((A005248(n)^2  4)/5).
a(n) = A007598(n)  A007598(n2), n > 1.
a(n) = (ap^n  am^n)/(apam), with ap := (3+sqrt(5))/2, am := (3sqrt(5))/2.
Invert transform of natural numbers: a(n) = Sum_{k=1..n} k*a(nk), a(0) = 1.  Vladeta Jovovic, Apr 27 2001
a(n) = S(n1, 3) with S(n, x) = U(n, x/2) Chebyshev's polynomials of the 2nd kind, see A049310.
a(n) = Sum_{k=0..n} binomial(n, k)*F(k).  Benoit Cloitre, Sep 03 2002
Limit_{n>infinity} a(n)/a(n1) = 1 + phi = (3 + sqrt(5))/2. This sequence includes all of the elements of A033888 combined with A033890.
a(0)=0, a(1)=1, a(2)=3, a(n)*a(n2) + 1 = a(n1)^2.  Benoit Cloitre, Dec 06 2002
a(n) = n + Sum_{k=0..n1} Sum_{i=0..k} a(i) = n + A054452(n).  Benoit Cloitre, Jan 26 2003
a(n) = Sum_{k=1..n} binomial(n+k1, nk).  Vladeta Jovovic, Mar 23 2003
E.g.f.: (2/sqrt(5))*exp(3*x/2)*sinh(sqrt(5)*x/2).  Paul Barry, Apr 11 2003
Second diagonal of array defined by T(i, 1) = T(1, j) = 1, T(i, j) = Max(T(i1, j) + T(i1, j1); T(i1, j1) + T(i, j1)).  Benoit Cloitre, Aug 05 2003
a(n) = F(n)*L(n) = A000045(n)*A000032(n).  Lekraj Beedassy, Nov 17 2003
F(2n+2) = 1, 3, 8, ... is the binomial transform of F(n+2).  Paul Barry, Apr 24 2004
Partial sums of A001519(n).  Lekraj Beedassy, Jun 11 2004
a(n) = Sum_{i=0..n1} binomial(2*n1i, i)*5^(ni1)*(1)^i.  Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n) = Sum_{k=0..n} binomial(n+k, nk1) = Sum_{k=0..n} binomial(n+k, 2k+1).
a(n+1) = Sum_{k=0..floor(n/2)} binomial(nk, k)*(1)^k*3^(n2*k).  Paul Barry, Oct 25 2004
a(n) = (n*L(n)  F(n))/5 = Sum_{k=0..n1} (1)^n*L(2*n2*k1).
The ith term of the sequence is the entry (1, 2) in the ith power of the 2 X 2 matrix M = ((1, 1), (1, 2)).  Simone Severini, Oct 15 2005
Computation suggests that this sequence is the Hankel transform of A005807. The Hankel transform of {a(n)} is Det[{{a(1), ..., a(n)}, {a(2), ..., a(n+1)}, ..., {a(n), ..., a(2n1)}}].  John W. Layman, Jul 21 2000
a(n+1) = (A005248(n+1)  A001519(n))/2.  Creighton Dement, Aug 15 2004
a(n+1) = Sum_{i=0..n} Sum_{j=0..n} binomial(ni, j)*binomial(nj, i).  N. J. A. Sloane, Feb 20 2005
a(n) = (2/sqrt(5))*sinh(2*n*psi), where psi:=log(phi) and phi=(1+sqrt(5))/2.  Hieronymus Fischer, Apr 24 2007
a(n) = ((phi+1)^n  A001519(n))/phi with phi=(1+sqrt(5))/2.  Reinhard Zumkeller, Nov 22 2007
Row sums of triangle A135871.  Gary W. Adamson, Dec 02 2007
a(n)^2 = Sum_{k=1..n} a(2*k1). This is a property of any sequence S(n) such that S(n) = B*S(n1)  S(n2) with S(0) = 0 and S(1) = 1 including {0,1,2,3,...} where B = 2.  Kenneth J Ramsey, Mar 23 2008
a(n) = 1/sqrt(5)*(phi^(2*n+2)  phi^(2*n2)), where phi = (1+sqrt(5))/2, the golden ratio.  Udita Katugampola (SIU), Sep 24 2008
If p[i] = i and if A is Hessenberg matrix of order n defined by: A[i,j] = p[ji+1], (i<=j), A[i,j] = 1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n) = det(A).  Milan Janjic, May 02 2010
If p[i] = Stirling2(i,2) and if A is the Hessenberg matrix of order n defined by: A[i,j] = p[ji+1], (i<=j), A[i,j] = 1, (i = j+1), and A[i,j] = 0 otherwise. Then, for n >= 1, a(n1) = det(A).  Milan Janjic, May 08 2010
a(n) = F(2*n+10) mod F(2*n+5).
a(n) = 1 + a(n1) + Sum_{i=1..n1} a(i), with a(0)=0.  Gary W. Adamson, Feb 19 2011
a(n) is equal to the permanent of the (n1) X (n1) Hessenberg matrix with 3's along the main diagonal, i's along the superdiagonal and the subdiagonal (i is the imaginary unit), and 0's everywhere else.  John M. Campbell, Jun 09 2011
a(n), n > 1 is equal to the determinant of an (nx) X (n1) tridiagonal matrix with 3's in the main diagonal, 1's in the super and subdiagonals, and the rest 0's.  Gary W. Adamson, Jun 27 2011
a(n) = b such that Integral_{x=0..Pi/2} sin(n*x)/(3/2cos(x)) dx = c + b*log(3).  Francesco Daddi, Aug 01 2011
a(n+1) = Sum_{k=0..n} A101950(n,k)*2^k.  Philippe Deléham, Feb 10 2012
G.f.: A(x) = x/(13*x+x^2) = G(0)/sqrt(5); where G(k)= 1 (a^k)/(1  b*x/(b*x  2*(a^k)/G(k+1))), a = (73*sqrt(5))/2, b = 3+sqrt(5), if x<(3sqrt(5))/2 = 0.3819660...; (continued fraction 3 kind, 3step ).  Sergei N. Gladkovskii, Jun 25 2012
a(n) = 2^n*b(n;1/2) = b(n;1), where b(n;d), n=0,1,...,d, denote the deltaFibonacci numbers defined in comments to A000045 (see also Witula's et al. papers).  Roman Witula, Jul 12 2012
Product_{n>=1} (1 + 1/a(n)) = 1 + sqrt(5).  Peter Bala, Dec 23 2012
Product_{n>=2} (1  1/a(n)) = (1/6)*(1 + sqrt(5)).  Peter Bala, Dec 23 2012
G.f.: x/(12*x) + x^2/(12*x)/(Q(0)x) where Q(k) = 1  x/(x*k+1)/Q(k+1); (continued fraction).  Sergei N. Gladkovskii, Feb 23 2013
G.f.: G(0)/2  1, where G(k) = 1 + 1/( 1  x/(x + (1x)^2/G(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Jul 16 2013
G.f.: x*G(0)/(23*x), where G(k) = 1 + 1/( 1  x*(5*k9)/(x*(5*k4)  6/G(k+1) )); (continued fraction).  Sergei N. Gladkovskii, Jul 17 2013
Sum_{n>=1} 1/(a(n) + 1/a(n)) = 1. Compare with A001519, A049660 and A049670.  Peter Bala, Nov 29 2013
a(n) = U(n1,3/2) where U(n1,x) is Chebyshev polynomial of the second kind.  Milan Janjic, Jan 25 2015
The o.g.f. A(x) satisfies A(x) + A(x) + 6*A(x)*A(x) = 0. The o.g.f. for A004187 equals A(sqrt(x))*A(sqrt(x)).  Peter Bala, Apr 02 2015
For n > 1, a(n) = (3*F(n+1)^2 + 2*F(n2)*F(n+1)  F(n2)^2)/4.  J. M. Bergot, Feb 16 2016
For n > 3, a(n) = floor(MA)  4 for n even and floor(MA) + 5 for n odd. MA is the maximum area of a quadrilateral with lengths of sides in order L(n), L(n), F(n3), F(n+3), with L(n)=A000032(n). The ratio of the longer diagonal to the shorter approaches 5/3.  J. M. Bergot, Feb 16 2016
a(n+1) = Sum_{j=0..n} Sum_{k=0..j} binomial(nj,k)*binomial(j,k)*2^(jk).  Tony Foster III, Sep 18 2017
a(n) = Sum_{k=0..n1} Sum_{i=0..n1} C(k+i,ki).  Wesley Ivan Hurt, Sep 21 2017
a(n) = Sum_{k=1..A000041(n)} A305309(n, k), n >= 1. Also row sums of triangle A078812. Wolfdieter Lang, May 31 2018
a(n) = H(2*n, 1, 1/2) for n > 0 where H(n, a, b) > hypergeom([a  n/2, b  n/2], [1  n], 4).  Peter Luschny, Sep 03 2019
Sum_{n>=1} 1/a(n) = A153386.  Amiram Eldar, Oct 04 2020
a(n) = A249450(n) + 2. Leo Tavares, Oct 10 2021
a(n) = 2/(sqrt(5)*tan(2*arctan(phi^(2*n)))), where phi = A001622 is the golden ratio.  Diego Rattaggi, Nov 21 2021
a(n) = sinh(2*n*arcsinh(1/2))/sqrt(5/4).  Peter Luschny, May 21 2022


EXAMPLE

G.f. = x + 3*x^2 + 8*x^3 + 21*x^4 + 55*x^5 + 144*x^6 + 377*x^7 + 987*x^8 + ...
a(3) = 8 because there are exactly 8 idempotent orderpreserving full transformations on a 3element chain, namely: (1,2,3)>(1,1,1),(1,2,3)>(2,2,2),(1,2,3)>(3,3,3),(1,2,3)>(1,1,3),(1,2,3)>(2,2,3),(1,2,3)>(1,2,2),(1,2,3)>(1,3,3),(1,2,3)>(1,2,3)mappings are coordinatewise.  Abdullahi Umar, Sep 08 2008


MAPLE

with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S, card > 0), S=Sequence(U, card > 1), U=Sequence(Z, card >0)}, unlabeled]: seq(count(SeqSeqSeqL, size=n+1), n=0..28); # Zerinvary Lajos, Apr 04 2009
H := (n, a, b) > hypergeom([a  n/2, b  n/2], [1  n], 4):
a := n > `if`(n = 0, 0, H(2*n, 1, 1/2)):
seq(simplify(a(n)), n=0..30); # Peter Luschny, Sep 03 2019


MATHEMATICA

f[n_] := Fibonacci[2n]; Array[f, 28, 0] (* or *)
LinearRecurrence[{3, 1}, {0, 1}, 28] (* Robert G. Wilson v, Jul 13 2011 *)
Take[Fibonacci[Range[0, 60]], {1, 1, 2}] (* Harvey P. Dale, May 23 2012 *)
Table[ ChebyshevU[n1, 3/2], {n, 0, 30}] (* JeanFrançois Alcover, Jan 25 2013, after Michael Somos *)
CoefficientList[Series[(x)/(1  3x + x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 10 2014 *)


PROG

(PARI) {a(n) = fibonacci(2*n)}; /* Michael Somos, Dec 06 2002 */
(PARI) {a(n) = subst( poltchebi(n+1)*4  poltchebi(n)*6, x, 3/2)/5}; /* Michael Somos, Dec 06 2002 */
(PARI) {a(n) = polchebyshev( n1, 2, 3/2)}; /* Michael Somos Jun 18 2011 */
(PARI) Vec(x/(13*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Oct 24 2012
(Sage) [lucas_number1(n, 3, 1) for n in range(27)] # Zerinvary Lajos, Jun 25 2008
(Sage) [fibonacci(2*n) for n in range(0, 28)] # Zerinvary Lajos, May 15 2009
(MuPAD) numlib::fibonacci(2*n) $ n = 0..35; // Zerinvary Lajos, May 09 2008
(Haskell)
a001906 n = a001906_list !! n
a001906_list =
0 : 1 : zipWith () (map (* 3) $ tail a001906_list) a001906_list
 Reinhard Zumkeller, Oct 03 2011
(Python)
def a(n, adict={0:0, 1:1}):
if n in adict:
return adict[n]
adict[n]=3*a(n1)  a(n2)
return adict[n] # David Nacin, Mar 04 2012
(Maxima) makelist(fib(2*n), n, 0, 30); /* Martin Ettl, Oct 21 2012 */
(Magma) [Fibonacci(2*n): n in [0..30]]; // Vincenzo Librandi, Sep 10 2014


CROSSREFS

Fibonacci A000045 = union of this sequence and A001519.
Cf. A000041, A000032, A001622, A048996, A052529, A055991, A078812, A305309, A058038 (pairwise products), A153386.
Inverse sequences A130259 and A130260.
Cf. A249450.
Cf. A033888, A033890.
Sequence in context: A027932 A084625 A088305 * A278613 A072632 A001671
Adjacent sequences: A001903 A001904 A001905 * A001907 A001908 A001909


KEYWORD

nonn,easy,nice,core


AUTHOR

N. J. A. Sloane


STATUS

approved



