A200069 - OEIS

%I #27 Jul 08 2024 21:42:50

%S 1,4,29,168,1049,6380,39157,239568,1467313,8983636,55009613,336825720,

%T 2062427849,12628445756,77325345061,473471175072,2899114186081,

%U 17751582020260,108694812500093,665549816263752,4075231827556217,24953074921653644,152790313444845397

%N a(n) = 4*a(n-1) + 13*a(n-2) for n>2, a(1)=1, a(2)=4.

%C De Moivres formula : a(n)=(r^n-s^n)/(r-s), for r>s gives sequences with integers if r and s are conjugates. With r=2+sqrt(17) and s=2-sqrt(17), a(n+1)/a(n) converges to 2+sqrt(17).

%H Reinhard Zumkeller, <a href="/A200069/b200069.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,13).

%F a(n) = ((2+sqrt(17))^n-(2-sqrt(17))^n)/(2*sqrt(17)).

%F G.f.: x/(1-4*x-13*x^2). - _Bruno Berselli_, Nov 15 2011

%e a(3) = 4*4+13*1 = 29.

%t LinearRecurrence[{4,13}, {1,4}, 50]

%o (Haskell)

%o a200069 n = a200069_list !! (n-1)

%o a200069_list = 1 : 4 : zipWith (+)

%o    (map (* 4) $ tail a200069_list) (map (* 13) a200069_list)

%o -- _Reinhard Zumkeller_, Nov 15 2011

%Y Cf. A041025.

%K nonn,easy

%O 1,2

%A _Sture Sjöstedt_, Nov 13 2011

%E More terms from _Bruno Berselli_, Nov 15 2011