A004004 a(n) = (3^(2*n+1) - 8*n - 3)/16.

0, 1, 14, 135, 1228, 11069, 99642, 896803, 8071256, 72641337, 653772070, 5883948671, 52955538084, 476599842805, 4289398585298, 38604587267739, 347441285409712, 3126971568687473, 28142744118187326, 253284697063686007, 2279562273573174140, 20516060462158567341

Offset: 0

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

The o.g.f. of this sequence enabled the analysis of A162008, A162009 and A162010. - Johannes W. Meijer, Jun 27 2009

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Fransen, Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k), Math. Comp., 37 (1981), 475-497.
• C. L. Mallows, Letter to N. J. A. Sloane, May 16 1973
• 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.
• G. Viennot, Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi, J. Combin. Theory, A 29 (1980), 121-133.
• Index entries for linear recurrences with constant coefficients, signature (11,-19,9).

Crossrefs

From Johannes W. Meijer, Jun 27 2009: (Start)

Equals the second right hand column of triangle A162005 divided by 2.

Cf. A162008, A162009, A162010, A162011 and A162014 [2*(1+3*z)].

(End)

Programs

Mathematica

LinearRecurrence[{11, -19, 9}, {0, 1, 14}, 100] (* G. C. Greubel, Jul 06 2016 *)
Table[(3^(2 n + 1) - 8 n - 3)/16, {n, 0, 24}] (* Michael De Vlieger, Jul 08 2016 *)

Formula

G.f.: -x*(1+3*x)/(9*x-1)/(x-1)^2. - Simon Plouffe in his 1992 dissertation.

a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). - Johannes W. Meijer, Jun 27 2009

a(n) = a(n-1) + (3^(2*n-1) - 1)/2. - Lechoslaw Ratajczak, Jul 06 2016

E.g.f.: (-3 - 8*x + 3*exp(8*x))*exp(x)/16. - Ilya Gutkovskiy, Jul 07 2016