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A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
(Formerly M1183)
5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

REFERENCES

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; https://depositonce.tu-berlin.de/bitstream/11303/5553/5/heldt_daniel.pdf

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..300

E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, sec. 8, arXiv:math/0307050 [math.CO], 2003.

M. D. McIlroy, The number of states of a dynamic storage system, Computer J., 25 (No. 3, 1982), 388-392.

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.

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

Michael Woltermann, Problem 1826, Mathematics Magazine, 83 (2010), 304-305.

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1).

FORMULA

G.f.: -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 sub diagonal. Let V = vector [1,0,0,0,...]. The sequence is generated as leftmost column of M*V iterates. - Gary W. Adamson, Jun 07 2011

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=1..28); # Alois P. Heinz, Sep 06 2008

MATHEMATICA

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

PROG

(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

CROSSREFS

Sequence in context: A048285 A051529 A230554 * A257519 A257387 A094286

Adjacent sequences:  A005204 A005205 A005206 * A005208 A005209 A005210

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Apr 04 2002

STATUS

approved

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Last modified June 28 16:56 EDT 2017. Contains 288839 sequences.