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a(n) = 2*Fibonacci(2*n+2).
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%I #73 Sep 08 2022 08:44:48

%S 2,6,16,42,110,288,754,1974,5168,13530,35422,92736,242786,635622,

%T 1664080,4356618,11405774,29860704,78176338,204668310,535828592,

%U 1402817466,3672623806,9615053952,25172538050,65902560198,172535142544

%N a(n) = 2*Fibonacci(2*n+2).

%C The pairs (x, y) = (a(n), a(n+1)) satisfy x^2 + y^2 = 3*x*y + 4. - _Michel Lagneau_, Feb 01 2014

%H Vincenzo Librandi, <a href="/A025169/b025169.txt">Table of n, a(n) for n = 0..200</a>

%H Hacène Belbachir, Soumeya Merwa Tebtoub, László Németh, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Nemeth/nemeth7.html">Ellipse Chains and Associated Sequences</a>, J. Int. Seq., Vol. 23 (2020), Article 20.8.5.

%H Mark W. Coffey, James L. Hindmarsh, Matthew C. Lettington, John Pryce, <a href="http://arxiv.org/abs/1502.03085">On Higher Dimensional Interlacing Fibonacci Sequences, Continued Fractions and Chebyshev Polynomials</a>, arXiv:1502.03085 [math.NT], 2015 (see p. 32).

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).

%F G.f.: 2/(1 - 3*x + x^2).

%F a(n) = 3*a(n-1) - a(n-2).

%F a(n) = 2*A001906(n+1).

%F a(n) = A111282(n+2). - _Reinhard Zumkeller_, Apr 08 2012

%F a(n) = Fibonacci(2*n+1) + Lucas(2*n+1). - _Bruno Berselli_, Oct 13 2017

%p seq( 2*fibonacci(2*n+2), n=0..30); # _G. C. Greubel_, Jan 16 2020

%t Table[2Fibonacci[2n+2], {n,0,30}] (* or *)

%t CoefficientList[Series[2/(1-3x+x^2), {x,0,30}], x] (* _Michael De Vlieger_, Mar 09 2016 *)

%t LinearRecurrence[{3, -1}, {2, 6}, 30] (* _Jean-François Alcover_, Sep 27 2017 *)

%o (PARI) a(n)=2*fibonacci(2*n+2)

%o (Magma) [2*Fibonacci(2*n+2): n in [0..30]]; // _Vincenzo Librandi_, Jul 11 2011

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 2/(1-3*x + x^2) )); // _Marius A. Burtea_, Jan 16 2020

%o (Haskell)

%o a025169 n = a025169_list !! n

%o a025169_list = 2 : 6 : zipWith (-) (map (* 3) $ tail a025169_list) a025169_list

%o -- _Reinhard Zumkeller_, Apr 08 2012

%o (Sage) [2*fibonacci(2*n+2) for n in (0..30)] # _G. C. Greubel_, Jan 16 2020

%o (GAP) List([0..30], n-> 2*Fibonacci(2*n+2) ); # _G. C. Greubel_, Jan 16 2020

%Y Cf. A000032, A000045, A001906, A002878, A122367.

%K nonn,easy

%O 0,1

%A _Wouter Meeussen_

%E Better description from _Michael Somos_