OFFSET
0,2
COMMENTS
Convolution of A001906(n), n >= 1 (even-indexed Fibonacci numbers) with itself.
A001787 and this sequence arise in counting ordered trees of height at most k where only the rightmost branch at the root actually achieves this height and the count is by the number of edges, with k = 3 for A001787 and k = 4 for this sequence.
Gives the number of 3412-avoiding permutations containing exactly one subsequence of type 321. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
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 = 0..1000
Jean-Luc Baril, Toufik Mansour, José L. Ramírez, and Mark Shattuck, Catalan words avoiding a pattern of length four, Univ. de Bourgogne (France, 2024). See p. 5.
Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 4.
Karl Dilcher and Larry Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, arXiv:2405.12024 [math.CO], 2024. See p. 10.
Rigoberto Flórez, Robinson Higuita, and Alexander Ramírez, The resultant, the discriminant, and the derivative of generalized Fibonacci polynomials, arXiv:1808.01264 [math.NT], 2018.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See p. 23.
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv preprint arXiv:1201.6243, 2012. - From N. J. A. Sloane, May 09 2012
Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
Eunjeong Lee, Mikiya Masuda, and Seonjeong Park, On Schubert varieties of complexity one, arXiv:2009.02125 [math.AT], 2020.
Valentin Ovsienko and Serge Tabachnikov, Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences, arXiv:1705.01623 [math.CO], 2017. See p. 9.
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
John Riordan, Notes to N. J. A. Sloane, Jul. 1968
John Riordan, Letter to N. J. A. Sloane, Sep 26 1980 with notes on the 1973 Handbook of Integer Sequences. Note that the sequences are identified by their N-numbers, not their A-numbers.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).
FORMULA
a(n) = (2*(2*n+1)*F(2*(n+1))+3*(n+1)*F(2*n+1))/5 with F(n) = A000045 (Fibonacci numbers).
a(n) = -a(-4-n) = ((4*n+2)*F(2*n) + (7*n+5)*F(2*n+1))/5 with F(n) = A000045 (Fibonacci numbers).
a(n) = (2*a(n-1) + (n+1)*F(2n+4))/3, where F(n) = A000045 (Fibonacci numbers). - Emeric Deutsch, Oct 08 2002
G.f.: 1/(1 - 3*x + x^2)^2.
a(n) = (Sum_{k=0..n} S(k, 3)*S(n-k, 3)), where S(n, x) = U(n, x/2) is the n-th Chebyshev polynomial of the 2nd kind, A049310. - Paul Barry, Nov 14 2003
a(n) = Sum_{k=1..n+1} F(2k)*F(2(n-k+2)), where F(k) is the k-th Fibonacci number. - Dan Daly (ddaly(AT)du.edu), Apr 24 2008
a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). - Vincenzo Librandi, Mar 14 2011
a(n) ~ (7 + 3*sqrt(5))*n*cos(n*arccos(3/2))/5. - Stefano Spezia, Mar 29 2022
From Peter Bala, Nov 05 2024: (Start)
a(n) = Sum_{k = 0..n} (n + 2*k + 1)*binomial(n+k, 2*k).
a(n) = (n+1) * hypergeom([-n, n+1, (n+3)/2], [1/2, (n+1)/2], -1/4).
Second-order recurrence: n*a(n) = 3*(n + 1)*a(n-1) - (n + 2)*a(n-2) with a(0) = 1 and a(1) = 6. (End)
MAPLE
A001871:=1/(z**2-3*z+1)**2; # Simon Plouffe in his 1992 dissertation
f:= gfun:-rectoproc({a(n)=6*a(n-1)-11*a(n-2)+6*a(n-3)-a(n-4),
a(0)=1, a(1)=6, a(2)=25, a(3)=90}, a(n), remember):
map(f, [$0..50]); # Robert Israel, May 05 2017
MATHEMATICA
CoefficientList[Series[1/(1-3x+x^2)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 10 2012 *)
PROG
(PARI) a(n)=((4*n+2)*fibonacci(2*n)+(7*n+5)*fibonacci(2*n+1))/5
(Magma) I:=[1, 6, 25, 90]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2012
(PARI) Vec(1/(1-3*x+x^2)^2 + O(x^100)) \\ Altug Alkan, Oct 31 2015
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
Additional comments from Wolfdieter Lang, Apr 07 2000
STATUS
approved