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A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
(Formerly M1183)
10
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

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.

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.

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 September 25 21:33 EDT 2017. Contains 292500 sequences.