A014717 a(n) = (F(n+1) + L(n))^2 where F(n) are the Fibonacci numbers (A000045) and L(n) are the Lucas numbers (A000032).

9, 4, 25, 49, 144, 361, 961, 2500, 6561, 17161, 44944, 117649, 308025, 806404, 2111209, 5527201, 14470416, 37884025, 99181681, 259660996, 679801329, 1779742969, 4659427600, 12198539809, 31936191849, 83610035716, 218893915321, 573071710225, 1500321215376

Offset

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3). - Colin Barker, Apr 23 2015

G.f.: (9 - 14*x - x^2)/ ((1+x)*(1-3*x+x^2)). - Colin Barker, Apr 23 2015

a(n) = A013655(n)^2. - Hartmut F. W. Hoft, Apr 24 2015

a(n) = (1/5)*(22*(-1)^n + 19*Fibonacci(2*n) + 23*Fibonacci(2*n-1)). - Ehren Metcalfe, Mar 26 2016

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

a(n) = 3*a(n-1) - a(n-2) + 22*(-1)^n. - Greg Dresden, May 18 2020

Mathematica

Table[(Fibonacci[n+1] + LucasL[n])^2, {n, 0, 30}] (* Michael De Vlieger, Apr 24 2015 *)

Prog

(PARI) lucas(n) = if(n==0, 2, fibonacci(2*n)/fibonacci(n))
a(n) = (fibonacci(n+1)+lucas(n))^2 \\ Colin Barker, Apr 24 2015
(PARI) Vec( (9-14*x-x^2)/((1+x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Apr 23 2015
(PARI) a(n) = (2*fibonacci(n+1)+fibonacci(n-1))^2
(Magma) [(Fibonacci(n+1) + Lucas(n))^2: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015

Crossrefs

Cf. A000032, A000045, A013655.

Sequence in context: A174679 A168077 A173536 * A104728 A058093 A164032

Adjacent sequences: A014714 A014715 A014716 * A014718 A014719 A014720

Keyword

nonn,easy

Author

Mohammad K. Azarian

Extensions

Name corrected by Colin Barker, Apr 24 2015

Status

approved