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%I A301699 #34 Sep 08 2022 08:46:20
%S A301699 0,1,2,8,26,94,330,1178,4186,14914,53098,189122,673530,2398834,
%T A301699 8543498,30428162,108371354,385970386,1374653610,4895901602,
%U A301699 17437011514,62102837746,221182535242,787753281218,2805624912090,9992381298706,35588393716202
%N A301699 Generating function = g(g(x)), where g(x) = g.f. of Jacobsthal numbers A001045.
%C A301699 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).
%C A301699 The Dira (2017) article contains on pages 851 and 852 several other sequences that could be added to the OEIS.
%H A301699 Vincenzo Librandi, <a href="/A301699/b301699.txt">Table of n, a(n) for n = 0..1000</a>
%H A301699 Oboifeng Dira, <a href="http://www.seams-bull-math.ynu.edu.cn/downloadfile.jsp?filemenu=_201706&amp;filename=07_41(6).pdf">A Note on Composition and Recursion</a>, Southeast Asian Bulletin of Mathematics (2017), Vol. 41, Issue 6, 849-853.
%H A301699 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,4,-6,-4).
%F A301699 G.f.: (-2*x^3-x^2+x)/(4*x^4+6*x^3-4*x^2-3*x+1).
%F A301699 a(n) = 3*a(n-1) + 4*a(n-2) - 6*a(n-3) - 4*a(n-4). - _Vincenzo Librandi_, Mar 30 2018
%p A301699 f:=proc(a,b) local t1;
%p A301699 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);
%p A301699 lprint(t1);
%p A301699 series(t1,x,50);
%p A301699 seriestolist(%);
%p A301699 end;
%p A301699 f(1,2);
%t A301699 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 *)
%o A301699 (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 *)
%Y A301699 Cf. A001045, A073371, A270863.
%K A301699 nonn,easy
%O A301699 0,3
%A A301699 _N. J. A. Sloane_, Mar 29 2018

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