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A001519
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a(n) = 3*a(n-1)-a(n-2), with a(0) = a(1) = 1.
(Formerly M1439 N0569)
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1, 1, 2, 5, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, 514229, 1346269, 3524578, 9227465, 24157817, 63245986, 165580141, 433494437, 1134903170, 2971215073, 7778742049, 20365011074, 53316291173, 139583862445
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| If the initial term is deleted, this is a bisection of the Fibonacci sequence A000045.
Number of ordered trees with n+1 edges and height at most 3 (height=number of edges on a maximal path starting at the root). Number of directed column-convex polyominoes of area n+1. Number of nondecreasing Dyck paths of length 2n+2. - Emeric Deutsch, Jul 11 2001
Terms for n>1 are the solutions to : 5x^2-4 is a square. - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002
a(1) = 1, a(n+1) = smallest Fibonacci number greater than the n-th partial sum. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 21 2002
The fractional part of tau*a(n) decreases monotonically to zero. - Benoit Cloitre (benoit7848c(AT)orange.fr), Feb 01 2003
n such that floor(phi^2*n^2)-floor(phi*n)^2 = 1 where phi=(1+sqrt(5))/2 - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 16 2003
Number of leftist horizontally convex polyominoes with area n+1.
Number of 31-avoiding words of length n on alphabet {1,2,3} which do not end in 3. (e.g. n=3, we have 111,112,121,122,132,211,212,221,222,232,321,322 and 332). See A028859. - Jon Perry, Aug 04 2003
Appears to give all solutions >1 to the equation : x^2=ceiling(x*r*floor(x/r)) where r=phi=(1+sqrt(5))/2. - Benoit Cloitre, Feb 24, 2004
a(1) = 1, a(2) = 2, then the least number such that the square of any term is just less than the geometric mean of its neighbors. a(n+1)*a(n-1)> a(n)^2. - Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Apr 06 2004
All positive integer solutions of Pell equation b(n)^2 - 5*a(n)^2 = -4 together with b(n)=A002878(n), n>=0. - Wolfdieter Lang, Aug 31 2004
a(n) = L(n,3), where L is defined as in A108299; see also A002878 for L(n,-3). - Reinhard Zumkeller, Jun 01 2005
Essentially same as Pisot sequence E(2,5).
Number of permutations of [n+1] avoiding 321 and 3412. E.g. a(3) = 13 because the permutations of [4] avoiding 321 and 3412 are: 1234, 2134, 1324, 1243, 3124, 2314, 2143, 1423, 1342, 4123, 3142, 2413, 2341. - Bridget Eileen Tenner (bridget(AT)math.mit.edu), Aug 15 2005
Number of 1324-avoiding circular permutations on [n+1].
A subset of the Markoff numbers (A002559). - Robert G. Wilson v, Oct 05 2005.
(x,y)=(a(n),a(n+1)) are the solutions of x/(yz)+y/(xz)+z/(xy)=3 with z=1. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 29 2001
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) = 1. - Herbert Kociemba (kociemba(AT)t-online.de), Jun 10 2004
With interpolated zeros, counts closed walks of length n at the start or end node of P_4. a(n) counts closed walks of length 2n at the start or end node of P_4. The sequence 0,1,0,2,0,5,.. counts walks of length n between the start and second node of P_4. - Paul Barry, Jan 26 2005
a(n) = number of ordered trees on n edges containing exactly one non-leaf vertex all of whose children are leaves (every ordered tree must contain at least one such vertex). For example, a(0) = 1 because the root of the tree with no edges is not considered to be a leaf and the condition "all children are leaves" is vacuously satisfied by the root and a(4) = 13 counts all 14 ordered trees on 4 edges (A000108) except (ignore dots)
|..|
.\/.
which has two such vertices. - David Callan (callan(AT)stat.wisc.edu), Mar 02 2005
Number of directed column-convex polyominoes of area n. Example: a(2)=2 because we have the 1 X 2 and the 2 X 1 rectangles. - Emeric Deutsch, Jul 31 2006
Same as the number of Kekule structures in polyphenanthrene in terms of the number of hexagons in extended (1,1)-nanotubes. See Table 1 on page 411 of I. Lukovits and D. Janezic. - Parthasarathy Nambi (PachaNambi(AT)yahoo.com), Aug 22 2006
Number of free generators of degree n of symmetric polynomials in 3-noncommuting variables. - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 24 2006
Inverse: With phi = (sqrt(5) + 1)/2, log_phi((sqrt(5) a(n) + sqrt(5 a(n)^2 - 4))/2) = n for n >= 1. - David W. Cantrell (DWCantrell(AT)sigmaxi.net), Feb 19 2007
Consider a teacher who teaches one student, then he finds he can teach two students while the original student learns to teach a student. And so on with every generation an individual can teach one more student then he could before. a(n) starting at a(2) gives the total number of new students/teachers (see program). - Ben Thurston (benthurston27(AT)yahoo.com), Apr 11 2007
The Diophantine equation a(n)=m has a solution (for m>=1) iff ceiling(arsinh(sqr(5)*m/2)/log(phi))) != ceiling(arcosh(sqr(5)*m/2)/log(phi))) where phi is the golden ratio. An equivalent condition is A130255(m)=A130256(m). - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 24 2007
a(n+1)= B^(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. 2=`0`, 5=`00`, 13=`000`,..., in Wythoff code.
The sequence starting (1, 2, 5, 13,...) = row sums of triangle A140068. - Gary W. Adamson, May 04 2008
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, May 04 2008
a(n) = number of partitions pi of [n] (in standard increasing form) such that Flatten[pi] is a (2-1-3)-avoiding permutation. Example: a(4)=13 counts all 15 partitions of [4] except 13/24 and 13/2/4. Here "standard increasing form" means the entries are increasing in each block and the blocks are arranged in increasing order of their first entries. Also number that avoid 3-1-2. - David Callan (callan(AT)stat.wisc.edu), Jul 22 2008
Equals row sums of triangle A152251 starting with offset 1. [From Gary W. Adamson, Nov 30 2008]
From Gary W. Adamson, Dec 27 2008: (Start)
Let P = the partial sum operator, A000012: (1; 1,1; 1,1,1;...) and A153463
= M, the partial sum & shift operator. It appears that beginning with any
randomly taken sequence S(n), iterates of the operations M * S(n), -> M * ANS,
-> P * ANS,...etc, (or starting with P) will rapidly converge upon a two-
sequence limit cycle of (1, 2, 5, 13, 34,...) and (1, 1, 3, 8, 21,...). (End)
Row sums of triangle A153342 = odd indexed Fibonacci numbers. [From Gary W. Adamson, Dec 24 2008]
a(n) = 0.25*(A153266(n) + A153267(n)), apart from initial terms [From Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 02 2009]
Sum_{n>=0} atan(1/a(n)) = (3/4)Pi [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 27 2009]
Syntactically similar to A001906 in the sense that both have the n-th term t(n) given by t(n) = 3t(n-1) + t(n-2), for n > 1. [From Kailasam Viswanathan Iyer (kvi(AT)nitt.edu), Apr 24 2009]
Summation of the squares of fibonacci numbers taken 2 at a time. Offset 1. a(3)=5. [From Al Hakanson (hawkuu(AT)gmail.com), May 27 2009]
Number of musical compositions of Rhythm-music over a time period of n-1 units. Example: a(4)=13; indeed, denoting by R a rest over a time period of 1 unit and by N[j] a note over a period of j units, we have (writing N for N[1]): NNN, NNR, NRN, RNN, NRR, RNR, RRN, RRR, N[2]R, RN[2], NN[2], N[2]N, N[3]) (see the J. Groh reference, pp. 43-48). [From Juergen K. Groh (juergen.groh(AT)lhsystems.com), Jan 17 2010]
From Gary W. Adamson, Feb 18 2010: (Start)
Given an infinite lower triangular matrix M with (1, 2, 3,...) in every
column but the leftmost column shifted upwards one row. Then (1, 2, 5,...)
= Lim_{n->inf} M^n. (Cf. A144257). (End)
As a fraction: 8/71 = 0.112676 or 98/9701 = 0.010102051334... (Fraction 9/71 or 99/9701 for sequence without initial term). 19/71 or 199/9701 for sequence in reverse. [From M. Dols (markdols99(AT)yahoo.com), May 18 2010]
Starting (1, 2, 5, 13,...) = sums of A179806 row terms. [From Gary W. Adamson, Jul 28 2010]
Starting (1, 2, 5, 13,...) = row sums of triangle A179745 [From Gary W. Adamson, Jul 25 2010]
For n>=1, a(n) is the number of compositions (ordered integer partitions) of 2n-1 into an odd number of odd parts. O.g.f.:(x-x^3)/(1-3x^2+x^4)= A(A(x))where A(x)=
1/(1-x)-1/(1-x^2)
For n>0, determinant of the n x n tridiagonal matrix with 1's in the super and subdiagonals, (1,3,3,3,...) in the main diagonal, and the rest zeros. - Gary W. Adamson, Jun 27 2011
The Hosoya index H(n) of the n-path graph P_n is given by H(2n-1) = 0 and H(2n) = a(n+1). [Eric W. Weisstein, Jul 11 2011]
The Gi3 sums, see A180662, of the triangles A108299 and A065941 equal the terms of this sequence without a(0). [Johannes W. Meijer, Aug 14 2011]
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REFERENCES
| 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.
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, Lecture Notes in Computer Science, No. 668, Springer, Berlin, 1993, pp. 282-298.
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 13,15.
S. Brlek, E. Duchi, E. Pergola and S. Rinaldi, On the equivalence problem for succession rules, Discr. Math., 298 (2005), 142-154.
N. G. de Bruijn, D. E. Knuth and S. O. Rice, The average height of planted plane trees, in: Graph Theory and Computing (ed. T. C. Read), Academic Press, New York, 1972, pp. 15-22.
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.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
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.
Jurgen Groh, Computerimprovisation mit Markoffketten und "kognitiven Algorithmen", Studienarbeit, Technische Hochschule Darmstadt, 1987. [From Juergen K. Groh (juergen.groh(AT)lhsystems.com), Jan 17 2010]
I. Lukovits and D. Janezic, "Enumeration of conjugated circuits in nanotubes", J. Chem. Inf. Comput. Sci., vol. 44 (2004) pp. 410-414.
J. G. Penaud and O. Roques, Generation de chemins de Dyck a pics croissants, Discrete Mathematics, Vol. 246, no. 1-3 (2002), 255-267.
S. Rinaldi and D. G. Rogers, Indecomposability: polyominoes and polyomino tilings, Math. Gaz., to appear, 2008.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 39.
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).
M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.
F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..200
E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos...
N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and coinvariants of the symmetric group in noncommuting variables, Canadian Journal of Mathematics 60:2 (2008), pp. 266-296.
D. Callan, Pattern avoidance in circular permutations.
David Callan, Pattern avoidance in "flattened" partitions .
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
Enrica Duchi, Andrea Frosini, Renzo Pinzani and Simone Rinaldi, A Note on Rational Succession Rules, J. Integer Seqs., Vol. 6, 2003.
J.-P. Ehrmann et al., POLYA003: Integers of the form a/(bc) + b/(ca) + c/(ab).
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 127
I. Jensen, Series exapansions for self-avoiding polygons
Tanya Khovanova, Recursive Sequences
R. Knott, Pi and the Fibonacci numbers [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Feb 27 2009]
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.
M. Renault, Dissertation
Eric Weisstein's World of Mathematics, Fibonacci Hyperbolic Functions
Eric Weisstein's World of Mathematics, Hosoya Index
D. Zeilberger, [math/9801016] Automated counting of LEGO towers
Index entries for sequences related to Chebyshev polynomials.
Index entries for sequences related to linear recurrences with constant coefficients
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FORMULA
| G.f.: (1-2*x)/(1-3*x+x^2).
a(n) = 3*a(n-1) - a(n-2).
a(n) = a(1-n).
a(1)=1, a(2)=2, a(n+2)=(a(n+1)^2+1)/a(n) - Benoit Cloitre, Aug 29 2002
a(n)=(phi^(2n-1)+phi^(1-2n))/sqrt(5) where phi=(1+sqrt(5))/2. - Michael Somos, Oct 28 2002
a(n) = A007598(n-1)+A007598(n) = A000045(n-1)^2+A000045(n)^2 = F(n)^2+F(n+1)^2 - Henry Bottomley (se16(AT)btinternet.com), Feb 09 2001
a(n)=sum(binomial(n+k, 2k), k=0..n). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001
a(n) ~ (1/5)*sqrt(5)*phi^(2*n+1) - Joe Keane (jgk(AT)jgk.org), May 15 2002
a(n) = Sum(k=0, n, C(n, k)*F(k+1)) - Benoit Cloitre, Sep 03 2002
Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then q(n, 1)=a(n) (this comment is essentially the same as that of L. Smiley) - Benoit Cloitre, Nov 10 2002
a(n) = (1/2)*(3*a(n-1)+sqrt(5*a(n-1)^2-4)) - Benoit Cloitre, Apr 12 2003
Main 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
Hankel transform of A002212. E.g. Det([1, 1, 3;1, 3, 10;3, 10, 36])= 5 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 25 2004
Solutions x > 0 to equation floor(x*r*floor(x/r)) = floor(x/r*floor(x*r)) when r=phi - Benoit Cloitre, Feb 15 2004
a(n)=sum(i=0, n, binomial(n+i, n-i)) - Jon Perry, Mar 08 2004
a(n)= S(n, 3) - S(n-1, 3) = T(2*n+1, sqrt(5)/2)/(sqrt(5)/2) with S(n, x)=U(n, x/2), resp. T(n, x), Chebyshev's polynomials of the second, resp. first kind. See triangle A049310, resp. A053120. - Wofdieter Lang, Aug 31 2004
a(n)= ((-1)^n)*S(2*n, I), with the imaginary unit I and S(n, x)=U(n, x/2) Chebyshev's polynomials of the second kind, A049310. - Wolfdieter Lang, Aug 31 2004
a(n)=sum_{0<=i_1<=i_2<=n}binomial(i_2, i_1)*binomial(n, i_1+i_2) - Benoit Cloitre, Oct 14 2004
a( n ) = a( n - 1 ) + SUM[ i = 0 to n - 1 ] a( i ) a( n ) = Fib( 2n + 1 ) SUM[ i = 0 to n - 1 ] a(i) = Fib( 2n ) - Andras Erszegi (erszegi.andras(AT)chello.hu), Jun 28 2005
The i-th term of the sequence is the entry (1, 1) of the i-th power of the 2 by 2 matrix M=((1, 1), (1, 2)). - Simone Severini, Oct 15 2005
a(n-1)=(1/n)*sum_{k=0...n}B(2k)*F(2n-2k)*binomial(2n, 2k) where B(2k) is the (2k)-th Bernoulli number - Benoit Cloitre, Nov 02 2005
a(n) = A055105(n,1)+A055105(n,2)+A055105(n,3) = A055106(n,1)+A055106(n,2) - Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 24 2006
a(n)=2/sqrt(5)*cosh((2n-1)*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 - phi*A001906(n) with phi=(1+sqrt(5))/2. - Reinhard Zumkeller, Nov 22 2007
a(n) = 2*a(n-1)+2*a(n-2)-a(n-3); a(n) = (sqrt(5.0) + 5.0)/10.0*(3.0/2.0 + sqrt(5.0)/2.0)^(n-2) + (-sqrt(5.0) + 5.0)/10.0*(3.0/2.0 - sqrt(5.0)/2.0)^(n-2). - Antonio A. Olivares (olivares14031(AT)yahoo.com), Mar 21 2008
a(n)=A147703(n,0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 29 2008]
a(n) = -a(n-1)+11*a(n-2)-4*a(n-3), this formula (it is one of two 3rd order linear recurrence relations given for this sequence) is a result of the generating floretion Z = X*Y with X = 1.5'i + 0.5i' + .25(ii + jj + kk + ee) and Y = 0.5'i + 1.5i' + .25(ii + jj + kk + ee). [From Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 02 2009]
With X,Y defined as X = ( F(n) F(n+1) ), Y = ( F(n+2) F(n+3) ), where F(n) is the n-th Fibonacci number (A000045), it follows a(n+2) = X.Y', where Y' is the transpose of Y (n >= 0). [From Kailasam Viswanathan Iyer, Apr 24 2009]
a(n)= Fibonacci(2n+2) mod Fibonacci(2n), n>1 [From Gary Detlefs, Nov 22 2010]
a(n) = (Fibonacci(n-1)^2+Fibonacci(n)^2+Fibonacci(2n-1))/2 [From Gary Detlefs, Nov 22 2010]
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EXAMPLE
| a(3)=13: there are 14 ordered trees with 4 edges; all of them, except for the path with 4 edges, have height at most 3.
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MAPLE
| A001519:=-(-1+z)/(1-3*z+z**2); [S. Plouffe in his 1992 dissertation. Gives sequence without an initial 1.]
A001519 := proc(n) option remember: if n=0 then 1 elif n=1 then 1 elif n>=2 then 3*procname(n-1)-procname(n-2) fi: end: seq(A001519(n), n=0..28); # [Johannes W. Meijer, Aug 14 2011]
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MATHEMATICA
| Fibonacci /@ (2Range[29] - 1) (from Robert G. Wilson v, Oct 05 2005)
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PROG
| (PARI) a(n)=fibonacci(2*n-1)
(PARI) a(n)=real(quadgen(5)^(2*n))
(PARI) a(n)=subst(poltchebi(n)+poltchebi(n-1), x, 3/2)*2/5
(Sage) [lucas_number1(n, 3, 1)-lucas_number1(n-1, 3, 1) for n in xrange(0, 30)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2009]
(Haskell)
a001519 n = a001519_list !! n
a001519_list = 1 : zipWith (-) (tail a001906_list) a001906_list
-- Reinhard Zumkeller, Jan 11 2012
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CROSSREFS
| Cf. A000045. First differences of A001906 and of A055588. a(n)= A060920(n, 0).
Row 3 of array A094954.
Equals A001654(n+1) - A001654(n-1), n>0.
Cf. A001653. A122367 is another version.
Cf. A055105, A055106, A055107, A074664, A124292, A124293, A124294, A124295, A140068, A140069, A153342, A153463.
Cf. inverse sequences A130255 and A130256. A152251 [From Gary W. Adamson, Nov 30 2008]
Cf. A153266, A153267 [From Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 02 2009]
Cf. A144257 [From Gary W. Adamson, Feb 18 2010]
Sequence in context: A141448 A011783 * A122367 A048575 A099496 A114299
Adjacent sequences: A001516 A001517 A001518 * A001520 A001521 A001522
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KEYWORD
| nonn,nice,easy,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Entry revised by N. J. A. Sloane (njas(AT)research.att.com), Aug 24 2006, May 13 2008
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