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A001906
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F(2*n) = bisection of Fibonacci sequence: a(n)=3*a(n-1)-a(n-2).
(Formerly M2741 N1101)
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278
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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
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0,3
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COMMENTS
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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.
a(n)=n+sum(k=0,n-1,sum(i=0,k,a(i))). - Benoit Cloitre, Jan 26 2003
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 second-order 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(i-1)| = 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
a(n+1) = (A005248(n+1) - A001519(n))/2. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Aug 15 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) = the number of unique 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 k-1 strictly increasing runs. Example: a(3)=F(6)=8 because we have 0|0,1|0,1|1,0|01,01|0,1|01,01|1 and 01|01. 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 DELEHAM, Apr 13 2007
The Diophantine equation a(n)=m has a solution (for m>=1) iff floor(arsinh(sqrt(5)*m/2)/ln(phi))<>floor(arcosh(sqrt(5)*m/2)/ln(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 order-preserving partial transformations (of an n-element chain) of width n (width(alpha) = max(Im(alpha))). Equivalently, it is the number of idempotent order-preserving full transformations (of an n-element chain). [From Abdullahi Umar, Sep 08 2008]
Contribution from Udita Katugampola (SIU) (uditanalin(AT)yahoo.com), Sep 24 2008: (Start)
a(n) is the number of ways that a string of 0,1 and 2 of size (n-1) can be arranged with no 12-pairs.
Here a(1)=3, a(2)=8, a(3)=21 and so on. a(n)=1/sqrt(5){phi^(2*n+2)-phi^(-2*n-2)}, where phi=(1+sqrt(5))/2 - The Golden Ratio. (End)
Starting with offset 1 = row sums of triangle A175011 [From Gary W. Adamson, Apr 03 2010]
As a fraction: 1/71 = 0.01408450... or 1/9701 = 0.0001030821.... [From M. Dols (markdols99(AT)yahoo.com), 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). [From 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..(n-2)/2} (1 + 4*Cos^2 k*Pi/n) = even indexed Fibonacci numbers with a(n) relating to the 2*n-Gons. The constants as products = roots to even indexed rows of triangle A152063. For example: a(5) = 55 satisfies the product formula relating to the 10-Gon. - [From Gary W. Adamson, Aug 15 2010]
Alternatively, product of roots to x^4 - 12x^3 + 51x^2 - 90x + 55, (10-th row of triangle A152063) = (4.618...)*(3.618...)*(2.381...)*(1.381...) = 55. - [From 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,...). [From Milan Janjic, Aug 26 2010]
Starting with "1" = row sums of triangle A180339, and eigensequence of triangle A137710. - [From 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
a(n) = 2^n b(n;1/2) = -b(n;-1), where b(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 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
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REFERENCES
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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.
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.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, [http://algo.inria.fr/banderier/Papers/DiscMath99.ps Generating Functions for Generating Trees], Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
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.
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.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Marc Chamberland and Christopher French, Generalized Catalan Numbers and Generalized Hankel Transformations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.1.
D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted 2012).
R. J. Douglas, Tournaments that admit exactly one Hamiltonian cycle, Proc. London Math. Soc., 21 (1970), 716-730.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
M. R. Garey, On enumerating tournaments that admit exactly one Hamiltonian circuit, J. Combin. Theory, B 13 (1972), 266-269.
A. Gerardin, Reply to Query 4389, L'Interm\'{e}diaire des Math\'{e}maticiens, 22 (1915), 23.
A. Gougenheim, About the linear sequence of integers such that each term is the sum of the two preceding, Fib. Quart., 9 (1971), 277-295, 298.
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case a=0,b=1; p=3, q=-1.
Howie, J. M. Combinatorial and probabilistic results in transformation semigroups. Words, languages and combinatorics, II (Kyoto, 1992), 200--206, World Sci. Publ., River Edge, NJ, (1994). [From Abdullahi Umar, Sep 08 2008]
M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. - From N. J. A. Sloane, Sep 16 2012
Kuba, Markus; Panholzer, Alois. Enumeration formulae for pattern restricted Stirling permutations. Discrete Math. 312 (2012), no. 21, 3179--3194. MR2957938. - From N. J. A. Sloane, Sep 25 2012
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq. (44) lhs, m=5.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From Abdullahi Umar, Sep 08 2008]
I. Lukovits, A. Graovac, E. Kalman, G. Kaptay, P. Nagy, S. Nikolic, J. Sytchev and N. Trinajstich, "Nanotubes: Number of Kekule Structures and Aromaticity", J. Chem. Inf. Comput. Sci, vol. 43 (2003), pp. 609-614. See Equation 6 on page 611.
G. Narang and A. K. Agarwal, Lattice paths and n-color compositions, Discr. Math., 308 (2008), 1732-1740.
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', Springer-Verlag 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, p. 96-100.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
A. Bremner and N. Tzanakis, Lucas sequences whose 12th or 9th term is a square
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
Howie, J. M. Products of idempotents in certain semigroups of transformations, Proc. Edinburgh Math. Soc. 17 (1971), 223-236. [From Abdullahi Umar, Sep 08 2008]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 147
Tanya Khovanova, Recursive Sequences
I. Lukovits and D. Janezic, Enumeration of conjugated circuits in nanotubes, J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.
_Simon Plouffe_, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology
N. J. A. Sloane, Transforms
M. Somos, In the Elliptic Realm.
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042
Index entries for sequences related to Chebyshev polynomials.
Index entries for two-way infinite sequences
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-1)
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FORMULA
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G.f.: x / (1 - 3*x + x^2).
a(n) = 3*a(n-1) - a(n-2).
a(n) = -a(-n).
a(n) = (ap^n - am^n)/(ap-am), with ap := (3+sqrt(5))/2, am := (3-sqrt(5))/2.
Invert transform of natural numbers: a(n)=Sum_{k=1..n} k*a(n-k), a(0)=1. - Vladeta Jovovic, Apr 27 2001
a(n) = S(n-1, 3) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the 2nd kind, see A049310.
a(n) = Sum(k=0, n, C(n, k)*F(k)) - Benoit Cloitre, Sep 03 2002
Lim. n-> Inf. a(n)/a(n-1) = 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(n-2)+1=a(n-1)^2. - Benoit Cloitre, Dec 06 2002
a(n) = Sum_{k=1..n} binomial(n+k-1, n-k). - 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(i-1, j)+T(i-1, j-1); T(i-1, j-1)+T(i, j-1)) - Benoit Cloitre, Aug 05 2003
a(n)=F(n)*L(n)=A000045(n)*A000032(n). - Lekraj Beedassy, Nov 17 2003
Fib(2n+2)=1, 3, 8, ... is the binomial transform of Fib(n+2). - Paul Barry, Apr 24 2004
Partial sums of A001519(n). - Lekraj Beedassy, Jun 11 2004
a(n)=Sum(C(2n-1-i, i)5^(n-i-1)(-1)^i, i=0, .., n-1). - Mario Catalani (mario.catalani(AT)unito.it), Jul 23 2004
a(n)=sum{k=0..n, binomial(n+k, n-k-1)}=sum{k=0..n, binomial(n+k, 2k+1)}
a(n+1)=sum{k=0..floor(n/2), binomial(n-k, k)(-1)^k*3^(n-2k)} - Paul Barry, Oct 25 2004
a(n) = (1/5)*( n*L(n)-F(n) ) = sum(k=0..n-1, (-1)^n*Lucas(2*n-2*k-1) ).
The i-th term of the sequence is the entry (1, 2) in the i-th power of the 2 by 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(2n-1)}}] - John W. Layman, Jul 21 2000
a(n+1) = Sum[i=0..n, Sum[j=0..n, C(n-i, j)*C(n-j, i) ]]. - N. J. A. Sloane, Feb 20 2005
a(n)=2/sqrt(5)*sinh(2n*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(2k-1) ). This is a property of any sequence S(n) such that S(n) = B*S(n-1) - S(n-2) 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*n-2)), where phi=(1+sqrt(5))/2 - The Golden Ratio [From Udita Katugampola (SIU) (uditanalin(AT)yahoo.com), Sep 24 2008]
If p[i]=i and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. [From 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[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)= det A. [From Milan Janjic, May 08 2010]
a(n) = fib(2n+10) mod fib(2n+5)
a(n) = 2*a(n-1) + a(n-2) + a(n-3) + ... + a(1) + 1, with a(0)=0. - Gary W. Adamson, Feb 19 2011
a(n) is equal to the permanent of the (n-1) X (n-1) 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. [From John M. Campbell, Jun 09 2011]
a(n), n>1 is equal to the determinant of an (n-x)x(n-1) tridiagonal matrix with 3's in the main diagonal, 1's in the super and subdiagonals, and the rest zeros. - Gary W. Adamson, Jun 27 2011
a(n)=b such that Integral_{x=0..Pi/2} sin(n*x)/(3/2-cos(x)) dx = c + b*ln(3) [From Francesco Daddi, Aug 01 2011]
a(n+1) = Sum_{k, 0<=k<=n} A101950(n,k)*2^k. - Philippe Deléham, Feb 10 2012
G.f. A(x)=x/(1-3*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=(7-3*sqrt(5))/2, b=3+sqrt(5), if |x|<(3-sqrt(5))/2=0.3819660....; (continued fraction 3 kind, 3-step ). - Sergei N. Gladkovskii, Jun 25 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/(1-2*x) + x^2/(1-2*x)/( Q(0) - x ) where Q(k) = 1 - x/(x*k + 1 )/Q(k+1) ; (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 23 2013
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EXAMPLE
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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 order-preserving full transformations on a 3-element 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 coordinate-wise. [From Abdullahi Umar, Sep 08 2008]
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MAPLE
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A001906:=1/(1-3*z+z**2); [Simon Plouffe in his 1992 dissertation.]
(Mupad) numlib::fibonacci(2*n) $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008
with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S, card > 0), S=Sequence(U, card > 1), U=Sequence(Z, card >0)}, unlabeled]: seq(count(SeqSeqSeqL, size=j), j=1..28); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2009]
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MATHEMATICA
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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[n-1, 3/2], {n, 0, 30}] (* Jean-François Alcover, Jan 25 2013, after Michael Somos *)
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PROG
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(PARI) {a(n) = fibonacci(2*n)} (PARI) Vec(x/(1-3*x+x^2)+O(x^99)) \\ Charles R Greathouse IV, Oct 24 2012
(PARI) {a(n) = subst( poltchebi(n+1)*4 - poltchebi(n)*6, x, 3/2)/5}
(PARI) {a(n) = polchebyshev( n-1, 2, 3/2)} \\ Michael Somos Jun 18 2011
(Sage) [lucas_number1(n, 3, 1) for n in range(27)] # Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008
(Sage) [fibonacci(2*n) for n in xrange(0, 28)] # From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009
(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(n-1) - a(n-2)
.return adict[n] # David Nacin, Mar 04 2012
(Maxima) makelist(fib(2*n), n, 0, 30); /* Martin Ettl, Oct 21 2012 */
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CROSSREFS
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a(n) = A060921(n-1, 0), n >= 1. Cf. A000045, A001519, A052529, A055991. a(n)=sqrt((A005248(n)^2-4)/5).
Apart from initial term, same as A088305.
Equals A007598(n) - A007598(n-2), n>1.
Second column of array A102310.and of A028412.
Cf. inverse sequences A130259 and A130260.
Sequence in context: A027932 A084625 A088305 * A072632 A001671 A090413
Adjacent sequences: A001903 A001904 A001905 * A001907 A001908 A001909
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KEYWORD
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nonn,easy,nice,core,changed
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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Lukovits et al. reference from Parthasarathy Nambi, Dec 21 2006
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STATUS
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approved
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