OFFSET
0,1
COMMENTS
The pairs (x, y) = (a(n), a(n+1)) satisfy x^2 + y^2 = 3*x*y + 4. - Michel Lagneau, Feb 01 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, Ellipse Chains and Associated Sequences, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.
Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials, arXiv:1502.03085 [math.NT], 2015 (see p. 32).
Tanya Khovanova, Recursive Sequences
Index entries for linear recurrences with constant coefficients, signature (3,-1).
FORMULA
G.f.: 2/(1 - 3*x + x^2).
a(n) = 3*a(n-1) - a(n-2).
a(n) = 2*A001906(n+1).
a(n) = A111282(n+2). - Reinhard Zumkeller, Apr 08 2012
a(n) = Fibonacci(2*n+1) + Lucas(2*n+1). - Bruno Berselli, Oct 13 2017
MAPLE
seq( 2*fibonacci(2*n+2), n=0..30); # G. C. Greubel, Jan 16 2020
MATHEMATICA
Table[2Fibonacci[2n+2], {n, 0, 30}] (* or *)
CoefficientList[Series[2/(1-3x+x^2), {x, 0, 30}], x] (* Michael De Vlieger, Mar 09 2016 *)
LinearRecurrence[{3, -1}, {2, 6}, 30] (* Jean-François Alcover, Sep 27 2017 *)
PROG
(PARI) a(n)=2*fibonacci(2*n+2)
(Magma) [2*Fibonacci(2*n+2): n in [0..30]]; // Vincenzo Librandi, Jul 11 2011
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 2/(1-3*x + x^2) )); // Marius A. Burtea, Jan 16 2020
(Haskell)
a025169 n = a025169_list !! n
a025169_list = 2 : 6 : zipWith (-) (map (* 3) $ tail a025169_list) a025169_list
-- Reinhard Zumkeller, Apr 08 2012
(Sage) [2*fibonacci(2*n+2) for n in (0..30)] # G. C. Greubel, Jan 16 2020
(GAP) List([0..30], n-> 2*Fibonacci(2*n+2) ); # G. C. Greubel, Jan 16 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Better description from Michael Somos
STATUS
approved