A004004 - OEIS

%I M4943 #48 Feb 08 2025 15:32:15

%S 0,1,14,135,1228,11069,99642,896803,8071256,72641337,653772070,

%T 5883948671,52955538084,476599842805,4289398585298,38604587267739,

%U 347441285409712,3126971568687473,28142744118187326,253284697063686007,2279562273573174140,20516060462158567341

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

%C The o.g.f. of this sequence enabled the analysis of A162008, A162009 and A162010. - _Johannes W. Meijer_, Jun 27 2009

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

%H G. C. Greubel, <a href="/A004004/b004004.txt">Table of n, a(n) for n = 0..1000</a>

%H A. Fransen, <a href="http://dx.doi.org/10.1090/S0025-5718-1981-0628708-X">Conjectures on the Taylor series expansion coefficients of the Jacobian elliptic function sn(n,k)</a>, Math. Comp., 37 (1981), 475-497.

%H C. L. Mallows, <a href="/A004004/a004004.pdf">Letter to N. J. A. Sloane, May 16 1973</a>

%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 G. Viennot, <a href="http://dx.doi.org/10.1016/0097-3165(80)90001-1">Une interprétation combinatoire des coefficients des développements en série entière des fonctions elliptiques de Jacobi</a>, J. Combin. Theory, A 29 (1980), 121-133.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-19,9).

%F G.f.: -x*(1+3*x)/(9*x-1)/(x-1)^2. - _Simon Plouffe_ in his 1992 dissertation.

%F a(n) = 11*a(n-1) - 19*a(n-2) + 9*a(n-3). - _Johannes W. Meijer_, Jun 27 2009

%F a(n) = a(n-1) + (3^(2*n-1) - 1)/2. - _Lechoslaw Ratajczak_, Jul 06 2016

%F E.g.f.: (-3 - 8*x + 3*exp(8*x))*exp(x)/16. - _Ilya Gutkovskiy_, Jul 07 2016

%t LinearRecurrence[{11, -19, 9}, {0, 1, 14}, 100] (* _G. C. Greubel_, Jul 06 2016 *)

%t Table[(3^(2 n + 1) - 8 n - 3)/16, {n, 0, 24}] (* _Michael De Vlieger_, Jul 08 2016 *)

%Y From _Johannes W. Meijer_, Jun 27 2009: (Start)

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

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

%Y (End)

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, _Simon Plouffe_