OFFSET
0,1
REFERENCES
Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
LINKS
Pridon Davlianidze, Problem B-1270, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 2 (2020), p. 179; Four Telescopic Infinite Products, Solution to Problem B-1270 by Jason L. Smith, ibid., Vol. 59, No. 2 (2021), pp. 183-184.
Emrah Kılıç, Yücel Türker Ulutaş, and Neşe Ömür, A Formula for the Generating Functions of Powers of Horadam's Sequence with Two Additional Parameters, J. Int. Seq. 14 (2011) #11.5.6, Table 2, k=2.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).
FORMULA
a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
a(n) = 2^(4*n)*(cos(Pi/5)^(2*n)+cos(3*Pi/5)^(2*n))^2. - Gary Detlefs, Dec 05 2010
From Gary Detlefs, Dec 06 2010: (Start)
a(n) = 7*a(n-1)-a(n-2)-10, n>1.
a(n) = 5*Sum_{k=0..n}(Fibonacci(4*k+2))+4, with offset -1. (End)
G.f.: -(9*x^2-23*x+4)/((x-1)*(x^2-7*x+1)). - Colin Barker, Jun 24 2012
Product_{n>=0} (1 + 5/a(n)) = 3*phi^2/2, where phi is the golden ratio (A001622) (Davlianidze, 2020). - Amiram Eldar, Dec 04 2024
MAPLE
luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d, `, luc(4*n)+2) od: # James A. Sellers, Mar 05 2003
G:=(x, n)-> cos(x)^n +cos(3*x)^n: seq(simplify(2^(4*n)*G(Pi/5, 2*n)^2), n=0..19) # Gary Detlefs, Dec 05 2010
t:= n-> sum(fibonacci(4*k+2), k=0..n):seq(5*t(n)+4, n=-1..18); # Gary Detlefs, Dec 06 2010
MATHEMATICA
LucasL[4*Range[0, 20]]+2 (* Harvey P. Dale, Sep 09 2012 *)
PROG
(Magma) [ Lucas(2*n)^2: n in [0..70] ]; // Vincenzo Librandi, Apr 16 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. K. Guy, Mar 04 2003
STATUS
approved