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A083579 Generalized Jacobsthal numbers.

%I A083579 #21 Sep 08 2022 08:45:10
%S A083579 0,1,1,4,8,19,39,82,166,337,677,1360,2724,5455,10915,21838,43682,
%T A083579 87373,174753,349516,699040,1398091,2796191,5592394,11184798,22369609,
%U A083579 44739229,89478472,178956956,357913927,715827867,1431655750,2863311514
%N A083579 Generalized Jacobsthal numbers.
%H A083579 G. C. Greubel, <a href="/A083579/b083579.txt">Table of n, a(n) for n = 0..1000</a>
%H A083579 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-3,2).
%F A083579 a(n) = (2^(n+3) - 5*(-1)^n - 3*(2*n+1))/12.
%F A083579 a(n+2) = a(n+1) + 2*a(n) + n, a(0)=0, a(1)=1.
%F A083579 G.f.: x*(1 - 2*x + 2*x^2)/(1 - 3*x + x^2 + 3*x^3 - 2*x^4). - _Colin Barker_, Jan 16 2012
%t A083579 LinearRecurrence[{3,-1,-3,2}, {0,1,1,4}, 40] (* _G. C. Greubel_, May 25 2019 *)
%o A083579 (PARI) concat(0, Vec(x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4) + O(x^40))) \\ _G. C. Greubel_, May 25 2019
%o A083579 (Magma) I:=[0,1,1,4]; [n le 4 select I[n] else 3*Self(n-1)-Self(n-2) - 3*Self(n-3)+2*Self(n-4): n in [1..40]]; // _G. C. Greubel_, May 25 2019
%o A083579 (Sage) (x*(1-2*x+2*x^2)/(1-3*x+x^2+3*x^3-2*x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 24 2019
%o A083579 (GAP) a:=[0,1,1,4];; for n in [5..40] do a[n]:=3*a[n-1]-a[n-2]-3*a[n-3] +2*a[n-4]; od; a; # _G. C. Greubel_, May 24 2019
%Y A083579 Cf. A083580.
%K A083579 nonn,easy
%O A083579 0,4
%A A083579 _Paul Barry_, May 01 2003