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%I M1183 #94 Nov 24 2023 14:55:34
%S 1,1,2,4,9,21,51,127,322,826,2135,5545,14445,37701,98514,257608,
%T 673933,1763581,4615823,12082291,31628466,82798926,216761547,
%U 567474769,1485645049,3889431721,10182603746,26658304492,69792188337,182718064101,478361686155,1252366480135
%N a(n) = (F(2*n-1) + F(n+1))/2 where F(n) is a Fibonacci number.
%C 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
%C a(n) = C(F(n+1)+1,2) + C(F(n)+1,2) = pairwise sums of A033192. - _Ralf Stephan_, Jul 06 2003
%C Number of (3412,54312)- and (3412,45321)-avoiding involutions in S_{n+1}. - _Ralf Stephan_, Jul 06 2003
%C 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
%C 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
%C 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
%C Convolution of [1,1,1,2,5,...], which is A001519 with another leading 1, and A212804. - _R. J. Mathar_, Apr 14 2018
%C a(n) is the number of Motzkin n-paths of height <= 3. - _Alois P. Heinz_, Nov 24 2023
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A005207/b005207.txt">Table of n, a(n) for n = 0..1000</a> (terms n = 1..300 from Vincenzo Librandi)
%H E. S. Egge, <a href="https://arxiv.org/abs/math/0307050">Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations</a>, sec. 8, arXiv:math/0307050 [math.CO], 2003.
%H S. Felsner, D. Heldt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Felsner/felsner2.html">Lattice Path Enumeration and Toeplitz Matrices</a>, J. Int. Seq. 18 (2015) # 15.1.3.
%H Daniel Heldt, <a href="https://depositonce.tu-berlin.de/bitstream/11303/5553/5/heldt_daniel.pdf">On the mixing time of the face flip-and up/down Markov chain for some families of graphs</a>, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
%H M. D. McIlroy, <a href="http://dx.doi.org/10.1093/comjnl/25.3.388">The number of states of a dynamic storage system</a>, Computer J., 25 (No. 3, 1982), 388-392.
%H M. D. McIlroy, <a href="/A005207/a005207.pdf">The number of states of a dynamic storage system</a>, Computer J., 25 (No. 3, 1982), 388-392. (Annotated scanned copy)
%H Heinrich Niederhausen, <a href="http://arxiv.org/abs/1105.3713">Inverses of Motzkin and Schroeder Paths</a>, arXiv preprint arXiv:1105.3713 [math.CO], 2011.
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H Michael Woltermann, <a href="http://www.jstor.org/stable/10.4169/mathmaga.83.4.0304a">Problem 1826</a>, Mathematics Magazine, 83 (2010), 304-305.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).
%F G.f.: 1-x*(1-2*x-x^2+x^3)/((x^2+x-1)*(x^2-3*x+1)).
%F a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4).
%F 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.
%F 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
%F a(-1-2*n) = A027994(2*n); a(-2*n)=A059512(2*n+1).
%F 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
%F 2*a(n) = A000045(n+1) + A001519(n). - _R. J. Mathar_, Apr 14 2018
%F a(n) mod 2 = A131719(n+3). - _Alois P. Heinz_, Nov 24 2023
%p 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
%p 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
%t LinearRecurrence[{4, -3, -2, 1}, {1, 2, 4, 9}, 30] (* _Jean-François Alcover_, Jan 31 2016 *)
%o (PARI) a(n)=(fibonacci(2*n-1)+fibonacci(n+1))/2
%o (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
%Y Cf. A000045, A001006, A001519, A131719.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
%E a(0)=1 prepended by _Alois P. Heinz_, Nov 24 2023
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