A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.

1, 1, 2, 4, 9, 21, 51, 127, 322, 826, 2135, 5545, 14445, 37701, 98514, 257608, 673933, 1763581, 4615823, 12082291, 31628466, 82798926, 216761547, 567474769, 1485645049, 3889431721, 10182603746, 26658304492, 69792188337, 182718064101, 478361686155, 1252366480135

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Number of block fountains with exactly n coins in the base when mirror image fountains are identified. - Michael Woltermann (mwoltermann(AT)washjeff.edu), Oct 06 2010

a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - Ralf Stephan, Jul 06 2003

Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - Ralf Stephan, Jul 06 2003

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. - Herbert Kociemba, May 31 2004

The sequence 1,1,2,4,9,... has g.f. 1/(1-x-x^2/(1-x-x^2/(1-x-x^2/(1-x))))=(1-3*x+x^2+x^2)/(1-4*x+3*x^2+2*x^3-x^4), and general term (A001519(n)+A000045(n+1))/2. It is the binomial transform of A001519 aerated. - Paul Barry, Dec 17 2009

The Kn3 and Kn4 sums, see A180662 for their definitions, of Losanitsch's triangle A034851 lead to this sequence. - Johannes W. Meijer, Jul 14 2011

Convolution of [1,1,1,2,5,...], which is A001519 with another leading 1, and A212804. - R. J. Mathar, Apr 14 2018

a(n) is the number of Motzkin n-paths of height <= 3. - Alois P. Heinz, Nov 24 2023

References

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (terms n = 1..300 from Vincenzo Librandi)
• E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8, arXiv:math/0307050 [math.CO], 2003.
• S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
• Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
• M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.
• M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy)
• Heinrich Niederhausen, Inverses of Motzkin and Schroeder Paths, arXiv preprint arXiv:1105.3713 [math.CO], 2011.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
• Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
• Michael Woltermann, Problem 1826, Mathematics Magazine, 83 (2010), 304-305.
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

Crossrefs

Cf. A000045, A001006, A001519, A131719.

Programs

Maple

A005207:=-(1-2*z-z^2+z^3)/(z^2-3*z+1)/(z^2+z-1); # Simon Plouffe in his 1992 dissertation with offset 0
a:= n-> (Matrix([[1,1,1,3]]). Matrix(4, (i,j)-> if i=j-1 then 1 elif j=1 then [4,-3,-2,1][i] else 0 fi)^n)[1,2]: seq(a(n), n=0..34); # Alois P. Heinz, Sep 06 2008

Mathematica

LinearRecurrence[{4, -3, -2, 1}, {1, 2, 4, 9}, 30] (* Jean-François Alcover, Jan 31 2016 *)

PARI

a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2

PARI

x='x+O('x^50); Vec(-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1))) \\ G. C. Greubel, Mar 05 2017

Formula

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

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).

a(n) = (w^(2*n-1) + w^(1-2*n) + w^(n+1) - (-w)^(-1-n))/(4*w-2) where w = (1+sqrt(5))/2.

a(n) = (2/5)*Sum_{k=1..4} ( sin(Pi*k/5)^2*(1 + 2*cos(Pi*k/5))^n ). - Herbert Kociemba, May 31 2004

a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).

Let M = an infinite tridiagonal matrix with all 1's in the super and main diagonals and [1,1,1,0,0,0,...] in the subdiagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. - Gary W. Adamson, Jun 07 2011

2*a(n) = A000045(n+1) + A001519(n). - R. J. Mathar, Apr 14 2018

a(n) mod 2 = A131719(n+3). - Alois P. Heinz, Nov 24 2023

Extensions

a(0)=1 prepended by Alois P. Heinz, Nov 24 2023