A014717 - OEIS

%I #47 Sep 08 2022 08:44:39

%S 9,4,25,49,144,361,961,2500,6561,17161,44944,117649,308025,806404,

%T 2111209,5527201,14470416,37884025,99181681,259660996,679801329,

%U 1779742969,4659427600,12198539809,31936191849,83610035716,218893915321,573071710225,1500321215376

%N a(n) = (F(n+1) + L(n))^2 where F(n) are the Fibonacci numbers (A000045) and L(n) are the Lucas numbers (A000032).

%H Colin Barker, <a href="/A014717/b014717.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 (2,2,-1).

%F a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3). - _Colin Barker_, Apr 23 2015

%F G.f.: (9 - 14*x - x^2)/ ((1+x)*(1-3*x+x^2)). - _Colin Barker_, Apr 23 2015

%F a(n) = A013655(n)^2. - _Hartmut F. W. Hoft_, Apr 24 2015

%F a(n) = (1/5)*(22*(-1)^n + 19*Fibonacci(2*n) + 23*Fibonacci(2*n-1)). - _Ehren Metcalfe_, Mar 26 2016

%F a(n) = (2^(-1-n)*(11*(-1)^n*2^(2+n) + (23-3*sqrt(5))*(3-sqrt(5))^n + (3+sqrt(5))^n*(23+3*sqrt(5))))/5. - _Colin Barker_, Oct 01 2016

%F a(n) = 3*a(n-1) - a(n-2) + 22*(-1)^n. - _Greg Dresden_, May 18 2020

%t Table[(Fibonacci[n+1] + LucasL[n])^2, {n, 0, 30}] (* _Michael De Vlieger_, Apr 24 2015 *)

%o (PARI) lucas(n) = if(n==0, 2, fibonacci(2*n)/fibonacci(n))

%o a(n) = (fibonacci(n+1)+lucas(n))^2 \\ _Colin Barker_, Apr 24 2015

%o (PARI) Vec( (9-14*x-x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ _Colin Barker_, Apr 23 2015

%o (PARI) a(n) = (2*fibonacci(n+1)+fibonacci(n-1))^2

%o (Magma) [(Fibonacci(n+1) + Lucas(n))^2: n in [0..30]]; // _Vincenzo Librandi_, Apr 25 2015

%Y Cf. A000032, A000045, A013655.

%K nonn,easy

%O 0,1

%A _Mohammad K. Azarian_

%E Name corrected by _Colin Barker_, Apr 24 2015