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%I #11 Sep 08 2022 08:45:23
%S 1,6,13,36,85,210,505,1224,2953,7134,17221,41580,100381,242346,585073,
%T 1412496,3410065,8232630,19875325,47983284,115841893,279667074,
%U 675176041,1630019160,3935214361,9500447886,22936110133,55372668156,133681446445,322735561050
%N Expansion of (1 +4*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
%C Elements of odd index give match to A075848: 2*n^2 + 9 is a square. Generating floretion: - 1.5'i + 'j + 'k - .5i' + j' + k' + .5'ii' - .5'jj' - .5'kk' - 'ij' + 'ik' - 'ji' + .5'jk' + 2'ki' - .5'kj' + .5e
%H Colin Barker, <a href="/A114689/b114689.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F G.f.: (1 +4*x -x^2)/((1-x)*(1+x)*(1-2*x-x^2)).
%F From _Colin Barker_, May 26 2016: (Start)
%F a(n) = (-1 - (-1)^n) + 3*((1+sqrt(2))^(1+n) - (1-sqrt(2))^(1+n))/(2*sqrt(2)).
%F a(n) = 2*a(n-1) +2*a(n-2) -2*a(n-3) -a(n-4) for n>3.
%F (End)
%F a(n) = 3*A000129(n+1) - (1 + (-1)^n). - _G. C. Greubel_, May 24 2021
%t Table[3*Fibonacci[n+1, 2] -1-(-1)^n, {n, 0, 30}] (* _G. C. Greubel_, May 24 2021 *)
%o (PARI) Vec((-1-4*x+x^2)/((1-x)*(x+1)*(x^2+2*x-1)) + O(x^30)) \\ _Colin Barker_, May 26 2016
%o (Magma) I:=[1,6,13,36]; [n le 4 select I[n] else 2*Self(n-1) +2*Self(n-2) -2*Self(n-3) -Self(n-4): n in [1..31]]; // _G. C. Greubel_, May 24 2021
%o (Sage) [3*lucas_number1(n+1,2,-1) -(1+(-1)^n) for n in (0..30)] # _G. C. Greubel_, May 24 2021
%Y Cf. A000129, A005409, A100828, A111954, A113224.
%Y Cf. A114647, A114688, A114695, A114696, A114697.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Feb 18 2006
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