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

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

Crossrefs

Cf. A000032, A000045, A013655.

Programs

Magma

[(Fibonacci(n+1) + Lucas(n))^2: n in [0..30]]; // Vincenzo Librandi, Apr 25 2015

Mathematica

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

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

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

Extensions

Name corrected by Colin Barker, Apr 24 2015