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A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.

%I A226308 #64 Nov 03 2024 15:14:06
%S A226308 2,1,5,10,17,37,74,145,293,586,1169,2341,4682,9361,18725,37450,74897,
%T A226308 149797,299594,599185,1198373,2396746,4793489,9586981,19173962,
%U A226308 38347921,76695845,153391690,306783377,613566757,1227133514,2454267025,4908534053,9817068106,19634136209
%N A226308 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0)=2, a(1)=1, a(2)=5.
%H A226308 Vincenzo Librandi, <a href="/A226308/b226308.txt">Table of n, a(n) for n = 0..1000</a>
%H A226308 G. Cerda-Morales, <a href="https://arxiv.org/abs/1905.00725">A note on modified third-order jacobsthal numbers</a>, arXiv:1905.00725 [math.CO], 2019.
%H A226308 Charles K. Cook and Michael R. Bacon, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_41_from27to39.pdf">Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations</a>, Annales Mathematicae et Informaticae, 41 (2013) pp. 27-39.
%H A226308 Evren Eyican Polatlı and Yüksel Soykan, <a href="https://doi.org/10.9734/ARJOM/2021/v17i230270">On generalized third-order Jacobsthal numbers</a>, Asian Res. J. of Math. (2021) Vol. 17, No. 2, 1-19, Article No. ARJOM.66022.
%H A226308 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A226308 G.f.: -(2*x^2-x+2) / ((2*x-1)*(x^2 + x + 1)). - _Colin Barker_, Jun 08 2013
%F A226308 a(3*n) = A047853(n+1), a(3*n+1) = A233328(n), a(3*n+2) = A046636(n+1). - _Philippe Deléham_, Feb 24 2014
%F A226308 From _Mélika Tebni_, Mar 09 2024: (Start)
%F A226308 E.g.f.: (1/7)*(8*exp(2*x) + exp(-x/2)*(6*cos(sqrt(3)*x/2) - 4*sqrt(3)*sin(sqrt(3)*x/2))) (Charles K. Cook and Michael R. Bacon, 2013).
%F A226308 a(n) = (1/7)*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)). (End)
%p A226308 A226308 := n -> 1/7*(2^(n+3) + 6*cos(2*Pi*n/3) - 4*sqrt(3)*sin(2*Pi*n/3)):
%p A226308 seq(A226308(n), n = 0 .. 34); # _Mélika Tebni_, Mar 09 2024
%t A226308 CoefficientList[Series[-(2 x^2 - x + 2) / ((2 x - 1) (x^2 + x + 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 18 2013 *)
%t A226308 LinearRecurrence[{1,1,2},{2,1,5},40] (* _Harvey P. Dale_, Nov 03 2024 *)
%o A226308 (Python)
%o A226308 a0, a1, a2 = 2, 1, 5
%o A226308 for n in range(77):
%o A226308   a = a2 + a1 + 2*a0
%o A226308   print(a0, end=', ')
%o A226308   a0, a1, a2 = a1, a2, a # _Alex Ratushnyak_, Jun 08 2013
%o A226308 (PARI) a(n)=([0,1,0; 0,0,1; 2,1,1]^n*[2;1;5])[1,1] \\ _Charles R Greathouse IV_, Jul 19 2016
%Y A226308 Cf. A046636, A047853, A233328.
%K A226308 nonn,easy
%O A226308 0,1
%A A226308 _N. J. A. Sloane_, Jun 07 2013
%E A226308 Deleted certain dangerous or potentially dangerous links. - _N. J. A. Sloane_, Jan 30 2021