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A001906 F(2*n) = bisection of Fibonacci sequence: a(n)=3*a(n-1)-a(n-2).
(Formerly M2741 N1101)
301
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

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, 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 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 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). - Abdullahi Umar, Sep 08 2008

From Udita Katugampola (SIU), 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. - Gary W. Adamson, Apr 03 2010

As a fraction: 1/71 = 0.01408450... or 1/9701 = 0.0001030821.... - 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). - 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. - 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. - 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

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

Solutions (x, y) = (a(n), a(n+1)) satisfying  x^2 + y^2 = 3xy + 1. - Michel Lagneau, Feb 01 2014

REFERENCES

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.

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.

D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers,  (submitted 2012).

A. Collins et al., Binary words, n-color compositions and ..., Fib. Quarterly, 51 (2013), 130-136.

Tomislav Doslic, Planar polycyclic graphs and their Tutte polynomials, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

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]

J. Jina and P. Trojovsky, On determinants of some tridiagonal matrices connected with Fibonacci numbers, International Journal of Pure and Applied Mathematics, Volume 88 No. 4 2013, 569-575; ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version); url: http://www.ijpam.eu; doi: http://dx.doi.org/10.12732/ijpam.v88i4.11

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.

T. Mansour, M. Shattuck, A statistic on n-color compositions and related sequences, Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 2, May 2014, pp. 127-140.

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.

LINKS

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

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.

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

Sergio Falcon, Catalan transform of the K-Fibonacci sequence, Commun. Korean Math. Soc. 28 (2013), No. 4, pp. 827-832; http://dx.doi.org/10.4134/CKMS.2013.28.4.827.

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

M. Janjic, Determinants and Recurrence Sequences, Journal of Integer Sequences, 2012, Article 12.3.5. - From N. J. A. Sloane, Sep 16 2012

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 séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology

J. Salas and A. D. Sokal, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial, J. Stat. Phys. 135 (2009) 279-373, arXiv:0711.1738. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014

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)

FORMULA

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 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(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. - 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[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. - 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. - 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. - 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*log(3). - 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); (continued fraction). - Sergei N. Gladkovskii, Feb 23 2013

G.f.: G(0)/2 - 1, where G(k)= 1 + 1/( 1 - x/(x + (1-x)^2/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 16 2013

G.f.: x*G(0)/(2-3*x), where G(k)= 1 + 1/( 1 - x*(5*k-9)/(x*(5*k-4) - 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

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 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. - Abdullahi Umar, Sep 08 2008

MAPLE

A001906:=1/(1-3*z+z**2); # Simon Plouffe in his 1992 dissertation

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); # Zerinvary Lajos, Apr 04 2009

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[n-1, 3/2], {n, 0, 30}] (* Jean-François Alcover, Jan 25 2013, after Michael Somos *)

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( n-1, 2, 3/2)}; /* Michael Somos Jun 18 2011 */

(PARI) Vec(x/(1-3*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 xrange(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(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 */

CROSSREFS

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

KEYWORD

nonn,easy,nice,core,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified July 31 13:31 EDT 2014. Contains 245085 sequences.