A060405 - OEIS

%I #34 Mar 09 2024 13:00:45

%S 2,2,5,9,19,40,88,198,455,1061,2501,5940,14182,33982,81625,196389,

%T 473039,1140260,2749988,6634458,16009555,38638441,93261961,225122760,

%U 543443402,1311905882,3167087405,7645809249,18458266699,44561632000

%N Sum of n-th Lucas number (A000032(n)) and n-th Pell number (A000129(n)).

%H Vincenzo Librandi, <a href="/A060405/b060405.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 (3,0,-3,-1).

%F From _Colin Barker_, Jun 22 2012: (Start)

%F a(n) = 3*a(n-1) - 3*a(n-3) - a(n-4).

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

%F a(n) = A000129(n) + A000032(n). - _Jonathan Vos Post_, Sep 02 2013

%e a(6) = Lucas(6) + Pell(6) = 18 + 70 = 88.

%p gfpell := x/(1-2*x-x^2): gfluc := (2-x)/(1-x-x^2): s := series(gfpell+gfluc, x, 100): for i from 0 to 60 do printf(`%d,`,coeff(s,x,i)) od:

%t LinearRecurrence[{3,0,-3,-1},{2,2,5,9},30] (* _Harvey P. Dale_, Jun 05 2017 *)

%o (Magma) I:=[2,2,5,9]; [n le 4 select I[n] else 3*Self(n-1)-3*Self(n-3)-Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Juan 07 2017

%Y Cf. A000032, A000129, A001932, A226638 Product of Pell and Lucas numbers.

%K nonn,easy

%O 0,1

%A _N. J. A. Sloane_, Apr 05 2001

%E More terms from _James A. Sellers_, Apr 06 2001