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A341905 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.

%I A341905 #18 Dec 18 2021 23:31:32
%S A341905 3,0,2,8,10,22,48,90,182,368,730,1462,2928,5850,11702,23408,46810,
%T A341905 93622,187248,374490,748982,1497968,2995930,5991862,11983728,23967450,
%U A341905 47934902,95869808,191739610,383479222,766958448,1533916890,3067833782,6135667568,12271335130
%N A341905 a(n) = a(n-1) + a(n-2) + 2*a(n-3) with a(0) = 3, a(1) = 0, a(2) = 2.
%H A341905 Michael De Vlieger, <a href="/A341905/b341905.txt">Table of n, a(n) for n = 0..3322</a>
%H A341905 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 A341905 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,2).
%F A341905 G.f.: (-3 + 3*x + x^2)/(-1 + x + x^2 + 2*x^3).
%F A341905 a(n) = (10*2^(n-1) + 13*A049347(n) - 9*A079978(n+1) + 3)/7. - _Greg Dresden_, Jun 20 2021
%p A341905 a:= n-> (<<0|1|0>, <0|0|1>, <2|1|1>>^n. <<3, 0, 2>>)[1,1]:
%p A341905 seq(a(n), n=0..34);  # _Alois P. Heinz_, Jun 04 2021
%t A341905 LinearRecurrence[{1, 1, 2}, {3, 0, 2}, 35] (* or *)
%t A341905 CoefficientList[Series[(-3 + 3 x + x^2)/(-1 + x + x^2 + 2 x^3), {x, 0, 34}], x]
%Y A341905 Cf. A001045, A077947, A186575, A226308.
%K A341905 nonn,easy
%O A341905 0,1
%A A341905 _Michael De Vlieger_, Jun 04 2021