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A005207 a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
(Formerly M1183)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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)
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.
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
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 *)
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
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Nov 24 2023
STATUS
approved

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Last modified March 28 12:26 EDT 2024. Contains 371254 sequences. (Running on oeis4.)