%I #26 Dec 31 2023 10:19:56
%S -1,3,5,13,31,75,181,437,1055,2547,6149,14845,35839,86523,208885,
%T 504293,1217471,2939235,7095941,17131117,41358175,99847467,241053109,
%U 581953685,1404960479,3391874643,8188709765,19769294173,47727298111,115223890395,278175078901,671574048197
%N a(n) = 2*a(n-1) + a(n-2), with a(0)= -1, a(1)= 3.
%C Double binomial transform of [1, 3, -5, 13, -31, 75, -181, ...] = the Pell-like sequence A048655: (1, 5, 11, 27, 65, 157, ...). - _Gary W. Adamson_, Jul 23 2008
%H G. C. Greubel, <a href="/A135532/b135532.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,1).
%F From _R. J. Mathar_, Feb 23 2008: (Start)
%F O.g.f.: (-1 + 5*x)/(1 - 2*x - x^2).
%F a(n) = 5*A000129(n) - A000129(n+1). (End)
%F a(n) = ((3+sqrt(2))*(1+sqrt(2))^n + (3-sqrt(2))*(1-sqrt(2))^n)/2 with offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Jun 17 2009
%F a(n) = (1/2)*(A002203(n) + 2*A002203(n-1)). - _G. C. Greubel_, May 22 2021
%t LinearRecurrence[{2,1},{-1,3},25] (* _G. C. Greubel_, Oct 17 2016 *)
%o (PARI) a(n)=([0,1; 1,2]^n*[-1;3])[1,1] \\ _Charles R Greathouse IV_, Oct 17 2016
%o (Magma) I:=[-1,3]; [n le 2 select I[n] else 2*Self(n-1) + Self(n-2): n in [1..30]]; // _G. C. Greubel_, May 22 2021
%o (Sage) [(lucas_number2(n,2,-1) + 2*lucas_number2(n-1,2,-1))/2 for n in (0..30)] # _G. C. Greubel_, May 22 2021
%Y Cf. A000129, A002203, A048655.
%K sign,easy
%O 0,2
%A _Paul Curtz_, Feb 21 2008
%E More terms from _R. J. Mathar_, Feb 23 2008