A001871 Expansion of 1/(1 - 3*x + x^2)^2. (Formerly M4166 N1733)

1, 6, 25, 90, 300, 954, 2939, 8850, 26195, 76500, 221016, 632916, 1799125, 5082270, 14279725, 39935214, 111228804, 308681550, 853904015, 2355364650, 6480104231, 17786356776, 48715278000, 133167004200, 363372003625, 989900286774

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

Matthew Blair, Rigoberto Flórez, and Antara Mukherjee, Honeycombs in the Pascal triangle and beyond, arXiv:2203.13205 [math.HO], 2022. See p. 4.

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 sequences related to Chebyshev polynomials.

Index entries for two-way infinite sequences

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) = ((4n+2)F(2n) + (7n+5)F(2n+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)) S(n, x) = U(n, x/2) Chebyshev polynomials of 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) = 2*A001870(n) - A238846(n). - Philippe Deléham, Mar 06 2014

a(n) ~ (7 + 3*sqrt(5))*n*cos(n*arccos(3/2))/5. - Stefano Spezia, Mar 29 2022

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

Partial sums of A001870 (one half of odd-indexed A001629(n), n >= 2, Fibonacci convolution).

Cf. A000045, A000051, A001629, A001820, A001906, A002720, A049310, A238846.

Sequence in context: A056279 A055337 A309946 * A000392 A365531 A099948

Adjacent sequences: A001868 A001869 A001870 * A001872 A001873 A001874

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Additional comments from Wolfdieter Lang, Apr 07 2000

Status

approved