A081069 Lucas(4n)+2, or Lucas(2n)^2.

4, 9, 49, 324, 2209, 15129, 103684, 710649, 4870849, 33385284, 228826129, 1568397609, 10749957124, 73681302249, 505019158609, 3461452808004, 23725150497409, 162614600673849, 1114577054219524, 7639424778862809

Offset

0,1

References

Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.

Links

Table of n, a(n) for n=0..19.

E. Kilic, Y. T. Ulutas, N. Omur, 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) = A005248(n)^2 = A056854(n)+2.

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

a(n) = 7*a(n-1)-a(n-2)-10, n>1. - Gary Detlefs, Dec 06 2010

a(n) = 5*sum(fibonacci(4*k+2),k=0..n)+4, with offset -1. - Gary Detlefs, Dec 06 2010

G.f.: -(9*x^2-23*x+4)/((x-1)*(x^2-7*x+1)). - Colin Barker, Jun 24 2012

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

Cf. A000032 (Lucas numbers).

Sequence in context: A231177 A110481 A030088 * A053967 A028945 A086541

Adjacent sequences: A081066 A081067 A081068 * A081070 A081071 A081072

Keyword

nonn,easy

Author

R. K. Guy, Mar 04 2003

Status

approved