A301699 Generating function = g(g(x)), where g(x) = g.f. of Jacobsthal numbers A001045.

0, 1, 2, 8, 26, 94, 330, 1178, 4186, 14914, 53098, 189122, 673530, 2398834, 8543498, 30428162, 108371354, 385970386, 1374653610, 4895901602, 17437011514, 62102837746, 221182535242, 787753281218, 2805624912090, 9992381298706, 35588393716202

Offset: 0

Author

N. J. A. Sloane, Mar 29 2018

Keywords

• nonn
• easy

Comments

The Dira (2017) article describes this as the self-convolution of A001045, but it is really the self-composition. - N. J. A. Sloane, Apr 07 2019, following a suggestion from Ilya Gutkovskiy. Note that A073371 is the convolution of A001045(n+1) with itself, with g.f.: g(x)^2/x^2, where g(x) = g.f. of A001045.

The Dira (2017) article contains on pages 851 and 852 several other sequences that could be added to the OEIS.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Oboifeng Dira, A Note on Composition and Recursion, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
• Index entries for linear recurrences with constant coefficients, signature (3,4,-6,-4).

Crossrefs

Cf. A001045, A073371, A270863.

Programs

Magma

I:=[0,1,2,8]; [n le 4 select I[n] else 3*Self(n-1)+4*Self(n-2)-6*Self(n-3)-4*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Mar 30 2018

Maple

f:=proc(a,b) local t1;
t1:=(x-a*x^2-b*x^3)/(1-3*a*x+(2*a^2-3*b)*x^2+3*a*b*x^3 + b^2*x^4);
lprint(t1);
series(t1,x,50);
seriestolist(%);
end;
f(1,2);

Mathematica

CoefficientList[Series[(-2 x^3 - x^2 + x) / (4 x^4 + 6 x^3 - 4 x^2 - 3 x + 1), {x, 0, 33}], x] (* Vincenzo Librandi, Mar 30 2018 *)

Formula

G.f.: (-2*x^3-x^2+x)/(4*x^4+6*x^3-4*x^2-3*x+1).

a(n) = 3*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4). - Vincenzo Librandi, Mar 30 2018