OFFSET
1,1
COMMENTS
Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6 etc.
The sequence satisfies the Pell equation a(n+1)^2 - 5*A081460(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..11
FORMULA
a(n) = a(n-1)^2 + 5*A081460(n-1)^2. - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
a(n) = (1/2)*(((4+2*sqrt(5))/2)^(2^(n-1)) + ((4-2*sqrt(5))/2)^(2^(n-1))). a(n+1) = 2*a(n)^2 - 1 for n > 1. - Artur Jasinski, Oct 12 2008
a(n) = A000032(3*2^(n-1))/2. - Ehren Metcalfe, Oct 05 2017
a(n) = A001077(2^(n-1)). - Robert FERREOL, Apr 16 2023
MATHEMATICA
k = 4; Table[Simplify[Expand[(1/2) (((k + Sqrt[k^2 + 4])/2)^(2^(n - 1)) + ((k - Sqrt[k^2 + 4])/2)^(2^(n - 1)))]], {n, 1, 6}] (* Artur Jasinski, Oct 12 2008 *)
aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (* Artur Jasinski, Oct 12 2008 *)
PROG
(PARI) {r=2; N=5; for(n=1, 8, a=numerator(r); b=denominator(r); print1(a, ", "); r=(1/2)*(r + N/r) )}
(Magma) m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Numerator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Mar 22 2003
EXTENSIONS
Edited and extended by Klaus Brockhaus and Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
STATUS
approved