A072110 - OEIS

%I #13 Sep 08 2022 08:45:06

%S 1,2,5,16,57,210,781,2912,10865,40546,151317,564720,2107561,7865522,

%T 29354525,109552576,408855777,1525870530,5694626341,21252634832,

%U 79315912985,296011017106,1104728155437,4122901604640,15386878263121

%N a(n) = 4*a(n-1) - a(n-2) - 2, with a(0)=1, a(1)=2.

%H G. C. Greubel, <a href="/A072110/b072110.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).

%F a(n) = A071954(n)/2 = A001353(n) + 1.

%F From _G. C. Greubel_, Feb 25 2019: (Start)

%F G.f.: (1-3*x)/((1-x)*(1-4*x+x^2))

%F a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3).

%F a(n) = 1 - (-i)^(n+1)*F(n, 4*i), where i = sqrt(-1) and F(n,x) is the Fibonacci polynomial. (End)

%t a[0]=1; a[1]=2; a[n_]:=a[n] =4*a[n-1]-a[n-2] -2; Table[a[n], {n, 0, 25}]

%t LinearRecurrence[{5,-5,1}, {1,2,5}, 30] (* _G. C. Greubel_, Feb 25 2019 *)

%o (Sage) [lucas_number1(n,4,1)+1 for n in range(26)] # _Zerinvary Lajos_, Jul 06 2008

%o (Sage) ((1-3*x)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 25 2019

%o (PARI) my(x='x+O('x^30)); Vec((1-3*x)/((1-x)*(1-4*x+x^2))) \\ _G. C. Greubel_, Feb 25 2019

%o (Magma) I:=[1,2,5]; [n le 3 select I[n] else 5*Self(n-1) -5*Self(n-2) + Self(n-3): n in [1..30]]; // _G. C. Greubel_, Feb 25 2019

%o (GAP) a:=[1,2,5];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # _G. C. Greubel_, Feb 25 2019

%Y Cf. A001353, A071954.

%K nonn

%O 0,2

%A _Robert G. Wilson v_, Jul 30 2002