%I A001906 M2741 N1101
%S A001906 0,1,3,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,
%T A001906 2178309,5702887,14930352,39088169,102334155,267914296,701408733,
%U A001906 1836311903,4807526976,12586269025,32951280099,86267571272
%N A001906 F(2n) = bisection of Fibonacci sequence: a(n)=3a(n-1)-a(n-2).
%C A001906 n such that 5*n^2 + 4 is a square. - Gregory V. Richardson (omomom(AT)hotmail.com),
Oct 13 2002
%C A001906 Apart from initial terms, also Pisot sequences E(3,8), P(3,8), T(3,8).
See A008776 for definitions of Pisot sequences.
%C A001906 a(n)=n+sum(k=0,n-1,sum(i=0,k,a(i))). - Benoit Cloitre (benoit7848c(AT)orange.fr),
Jan 26 2003
%C A001906 Binomial transform of A000045. - Paul Barry (pbarry(AT)wit.ie), Apr 11
2003
%C A001906 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
%C A001906 Simplest example of a second-order recurrence with the sixth term a square.
%C A001906 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
%C A001906 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
%C A001906 a(n+1) = (A005248(n+1) - A001519(n))/2. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de),
Aug 15 2004
%C A001906 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
%C A001906 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
%C A001906 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
%C A001906 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
%C A001906 See Table 1 on page 411 of Lukovits and Janezic paper. - Parthasarathy
Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
%C A001906 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
%C A001906 [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
%C A001906 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
%C A001906 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.
%C A001906 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
%C A001906 Equals row sums of triangles A140069, A140736 and A140737. - Gary W.
Adamson (qntmpkt(AT)yahoo.com), May 25 2008
%C A001906 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]
%C A001906 Contribution from Udita Katugampola (SIU) (uditanalin(AT)yahoo.com),
Sep 24 2008: (Start)
%C A001906 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.
%C A001906 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
%C A001906 Thus it gives every other Fibonacci number starting with 3. (End)
%D A001906 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences,
Academic Press, 1995 (includes this sequence).
%D A001906 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973
(includes this sequence).
%D A001906 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.
%D A001906 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.
%D A001906 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 A001906 Marc Chamberland and Christopher French, Generalized Catalan Numbers
and Generalized Hankel Transformations, Journal of Integer Sequences,
Vol. 10 (2007), Article 07.1.1.
%D A001906 R. J. Douglas, Tournaments that admit exactly one Hamiltonian cycle,
Proc. London Math. Soc., 21 (1970), 716-730.
%D A001906 G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence
Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
%D A001906 M. R. Garey, On enumerating tournaments that admit exactly one Hamiltonian
circuit, J. Combin. Theory, B 13 (1972), 266-269.
%D A001906 A. Gerardin, Reply to Query 4389, L'Interm\'{e}diaire des Math\'{e}maticiens,
22 (1915), 23.
%D A001906 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.
%D A001906 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.
%D A001906 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.
%D A001906 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.
%D A001906 I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes",
J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.
%D A001906 G. Narang and A. K. Agarwal, Lattice paths and n-color compositions,
Discr. Math., 308 (2008), 1732-1740.
%D A001906 I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers.
2nd ed., Wiley, NY, 1966, p. 101.
%D A001906 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]
%D A001906 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]
%D A001906 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]
%H A001906 T. D. Noe, <a href="b001906.txt">Table of n, a(n) for n=0..200</a>
%H A001906 <a href="Sindx_Tu.html#2wis">Index entries for two-way infinite sequences</
a>
%H A001906 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to
linear recurrences with constant coefficients</a>
%H A001906 A. Bremner and N. Tzanakis, <a href="http://arXiv.org/abs/math.NT/0405306">
Lucas sequences whose 12th or 9th term is a square</a>
%H A001906 Aleksandar Cvetkovic, Predrag Rajkovic and Milos Ivkovic, <a href="http:/
/www.cs.uwaterloo.ca/journals/JIS/index.html">Catalan Numbers, the
Hankel Transform and Fibonacci Numbers</a>, Journal of Integer Sequences,
Vol. 5 (2002), Article 02.1.3
%H A001906 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=147">
Encyclopedia of Combinatorial Structures 147</a>
%H A001906 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
RecursiveSequences.html">Recursive Sequences</a>
%H A001906 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al,
1992.
%H A001906 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
1031 Generating Functions and Conjectures</a>, Universit\'{e} du
Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A001906 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%H A001906 M. Somos, <a href="http://grail.cba.csuohio.edu/~somos/step2.txt">In
the Elliptic Realm</a>.
%H A001906 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
FibonacciHyperbolicFunctions.html">Fibonacci Hyperbolic Functions</
a>
%H A001906 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to
Chebyshev polynomials.</a>
%F A001906 G.f.: x/(1-3x+x^2).
%F A001906 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
%F A001906 a(n) = (ap^n - am^n)/(ap-am), with ap := (3+sqrt(5))/2, am := (3-sqrt(5))/
2.
%F A001906 a(n) = S(n-1, 3) with S(n, x)=U(n, x/2) Chebyshev's polynomials of the
2nd kind, see A049310.
%F A001906 a(n) = Sum(k=0, n, C(n, k)*F(k)) - Benoit Cloitre (benoit7848c(AT)orange.fr),
Sep 03 2002
%F A001906 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.
%F A001906 a(n) = 3*a(n-1) - a(n-2).
%F A001906 a(0)=0, a(1)=1, a(2)=3, a(n)a(n-2)+1=a(n-1)^2. - Benoit Cloitre, Dec
06 2002
%F A001906 a(n) = Sum_{k=1..n} binomial(n+k-1, n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs),
Mar 23 2003
%F A001906 E.g.f. (2/sqrt(5))exp(3x/2)sinh(sqrt(5)x/2) - Paul Barry (pbarry(AT)wit.ie),
Apr 11 2003
%F A001906 a(n) = 3*a(n-1) - a(n-2) = -a(-n).
%F A001906 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
%F A001906 a(n)=F(n)*L(n)=A000045(n)*A000032(n). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Nov 17 2003
%F A001906 Fib(2n+2)=1, 3, 8, ... is the binomial transform of Fib(n+2). - Paul
Barry (pbarry(AT)wit.ie), Apr 24 2004
%F A001906 Partial sums of A001519(n). - Lekraj Beedassy (blekraj(AT)yahoo.com),
Jun 11 2004
%F A001906 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
%F A001906 a(n)=sum{k=0..n, binomial(n+k, n-k-1)}=sum{k=0..n, binomial(n+k, 2k+1)}
%F A001906 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
%F A001906 a(n) = (1/5) [ n*L(n)-F(n) ] = Sum[k=0..n-1, (-1)^n*Lucas(2n-2k-1) ].
%F A001906 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
%F A001906 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
%F A001906 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
%F A001906 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
%F A001906 a(n) = ((phi+1)^n - A001519(n))/phi with phi=(1+sqrt(5))/2. - Reinhard
Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 22 2007
%F A001906 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
%F A001906 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
%F A001906 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]
%e A001906 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]
%p A001906 A001906:=1/(1-3*z+z**2); [S. Plouffe in his 1992 dissertation.]
%p A001906 (Mupad) numlib::fibonacci(2*n) $ n = 0..35; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
May 09 2008
%p A001906 with(combinat):with(finance):seq(fibonacci(futurevalue(n,1,1)),n=0..36);
- Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 02 2008
%p A001906 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]
%t A001906 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]
%o A001906 (PARI) a(n)=fibonacci(2*n)
%o A001906 (PARI) a(n)=subst(poltchebi(n+1)*4-poltchebi(n)*6,x,3/2)/5
%o A001906 sage: [lucas_number1(n,3,1) for n in range(27)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com),
Jun 25 2008
%o A001906 (Other) sage: [fibonacci(2*n) for n in xrange(0, 28)]# [From Zerinvary
Lajos (zerinvarylajos(AT)yahoo.com), May 15 2009]
%Y A001906 a(n) = A060921(n-1, 0), n >= 1. Cf. A000045, A001519, A052529, A055991.
a(n)=sqrt((A005248(n)^2-4)/5).
%Y A001906 Apart from initial term, same as A088305.
%Y A001906 Equals A007598(n) - A007598(n-2), n>1.
%Y A001906 Second column of array A102310.
%Y A001906 Second column of array A028412.
%Y A001906 Cf. inverse sequences A130259 and A130260.
%Y A001906 Cf. A135871.
%Y A001906 Cf. A130736, A130737, A140068, A140069, A140736.
%Y A001906 Sequence in context: A005580 A027932 A084625 this_sequence A088305 A072632
A001671
%Y A001906 Adjacent sequences: A001903 A001904 A001905 this_sequence A001907 A001908
A001909
%K A001906 nonn,easy,nice,core
%O A001906 0,3
%A A001906 N. J. A. Sloane (njas(AT)research.att.com).
%E A001906 Lukovits et al. reference from Parthasarathy Nambi (PachaNambi(AT)yahoo.com),
Dec 21 2006
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