0,3

n such that 5*n^2 + 4 is a square. - Gregory V. Richardson (omomom(AT)hotmail.com), 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 (benoit7848c(AT)orange.fr), Jan 26 2003

Binomial transform of A000045. - Paul Barry (pbarry(AT)wit.ie), 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 (deutsch(AT)duke.poly.edu), 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 (blekraj(AT)yahoo.com), 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 (andrew(AT)weimholt.com), 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. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), 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 (pbarry(AT)wit.ie), 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 (maxale(AT)gmail.com), 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 (deutsch(AT)duke.poly.edu), Jul 23 2006

See Table 1 on page 411 of Lukovits and Janezic paper. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), 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 (kolotoko(AT)wanadoo.fr), Apr 13 2007

The Diophantine equation a(n)=m has a solution (for m>=1) iff floor(arsinh(sqr(5)*m/2)/ln(phi))<>floor(arcosh(sqr(5)*m/2)/ln(phi)) where phi is the golden ratio. An equivalent condition is A130259(m)=A130260(m). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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.

Bisection of the Fibonacci sequence into odd indexed non-zero terms (1, 2, 5, 13,...) and even indexed terms (1, 3, 8, 21,...) may be represented as row sums of companion triangles A140068 and A140069. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2008

Equals row sums of triangles A140069, A140736 and A140737. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 A. Umar (aumarh(AT)squ.edu.om), 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 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

Thus it gives every other Fibonacci number starting with 3. (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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.

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.

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.

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.

I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.

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.

Howie, J. M. Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17 (1971), 223-236. [From A. Umar (aumarh(AT)squ.edu.om), Sep 08 2008]

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 A. Umar (aumarh(AT)squ.edu.om), Sep 08 2008]

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From A. Umar (aumarh(AT)squ.edu.om), Sep 08 2008]

T. D. Noe, Table of n, a(n) for n=0..200

Index entries for two-way infinite sequences

Index entries for sequences related to linear recurrences with constant coefficients

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 147

Tanya Khovanova, Recursive Sequences

S. 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.

S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.

N. J. A. Sloane, Transforms

M. Somos, In the Elliptic Realm.

Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions

Index entries for sequences related to Chebyshev polynomials.

G.f.: x/(1-3x+x^2).

Invert transform of natural numbers: a(n)=Sum_{k=1..n} k*a(n-k), a(0)=1. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 27 2001

a(n) = (ap^n - am^n)/(ap-am), with ap := (3+sqrt(5))/2, am := (3-sqrt(5))/2.

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 (benoit7848c(AT)orange.fr), 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(n) = 3*a(n-1) - a(n-2).

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 (vladeta(AT)eunet.rs), Mar 23 2003

E.g.f. (2/sqrt(5))exp(3x/2)sinh(sqrt(5)x/2) - Paul Barry (pbarry(AT)wit.ie), Apr 11 2003

a(n) = 3*a(n-1) - a(n-2) = -a(-n).

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 (benoit7848c(AT)orange.fr), Aug 05 2003

a(n)=F(n)*L(n)=A000045(n)*A000032(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), Nov 17 2003

Fib(2n+2)=1, 3, 8, ... is the binomial transform of Fib(n+2). - Paul Barry (pbarry(AT)wit.ie), Apr 24 2004

Partial sums of A001519(n). - Lekraj Beedassy (blekraj(AT)yahoo.com), 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 (pbarry(AT)wit.ie), Oct 25 2004

a(n) = (1/5) [ n*L(n)-F(n) ] = Sum[k=0..n-1, (-1)^n*Lucas(2n-2k-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 (ss54(AT)york.ac.uk), 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 (layman(AT)math.vt.edu), Jul 21 2000

Let M = {{0, -1}, {1, 3}}, v[n]=M.v[n-1]; then a(n) = Abs[v[n][[i]]. - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), May 29 2005

a(n)=2/sqrt(5)*sinh(2n*psi), where psi:=ln(phi) and phi=(1+sqrt(5))/2. - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Apr 24 2007

a(n) = ((phi+1)^n - A001519(n))/phi with phi=(1+sqrt(5))/2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 22 2007

Row sums of triangle A135871 such that F(2n) = sum of n-th row terms of A135871. Example: F(10) = 55 = (1 - 27 + 81). - Gary W. Adamson (qntmpkt(AT)yahoo.com), 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 (Ramsey2879(AT)msn.com), 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]

a(3)= 8 because the 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 A. Umar (aumarh(AT)squ.edu.om), Sep 08 2008]

A001906:=1/(1-3*z+z**2); [S. Plouffe in his 1992 dissertation.]

(Mupad) numlib::fibonacci(2*n) $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 09 2008

with(combinat):with(finance):seq(fibonacci(futurevalue(n, 1, 1)), n=0..36); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 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]

a[n_]:=Fibonacci[n]; lst={}; k=0; Do[If[k==0, AppendTo[lst, a[n]]]; If[k==0, k=1, k=0], {n, 0, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 13 2008]

(PARI) a(n)=fibonacci(2*n)

(PARI) a(n)=subst(poltchebi(n+1)*4-poltchebi(n)*6, x, 3/2)/5

sage: [lucas_number1(n, 3, 1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 25 2008

(Other) sage: [fibonacci(2*n) for n in xrange(0, 28)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]

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.

Second column of array A028412.

Cf. inverse sequences A130259 and A130260.

Cf. A135871.

Cf. A130736, A130737, A140068, A140069, A140736.

Sequence in context: A005580 A027932 A084625 this_sequence A088305 A072632 A001671

Adjacent sequences: A001903 A001904 A001905 this_sequence A001907 A001908 A001909

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

Lukovits et al. reference from Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Dec 21 2006