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A052995 Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).

%I A052995 #84 Jul 02 2025 16:01:58
%S A052995 0,2,4,10,26,68,178,466,1220,3194,8362,21892,57314,150050,392836,
%T A052995 1028458,2692538,7049156,18454930,48315634,126491972,331160282,
%U A052995 866988874,2269806340,5942430146,15557484098,40730022148,106632582346,279167724890,730870592324
%N A052995 Expansion of 2*x*(1 - x)/(1 - 3*x + x^2).
%C A052995 Terms >=4 give solutions x to floor(phi^2*x^2) - floor(phi*x)^2 = 5, where phi =(1 + sqrt(5))/2. - _Benoit Cloitre_, Mar 16 2003
%C A052995 Except for the first term, positive values of x (or y) satisfying x^2 - 18*x*y + y^2 + 256 = 0. - _Colin Barker_, Feb 14 2014
%C A052995 a(n+1) is the square of the distance AB, where A is the point (F(n), F(n+1)), B is the 90-degree rotation of A about the origin, and F(n)=A000045(n) are the Fibonacci numbers. - _Burak Muslu_, Mar 24 2021
%D A052995 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.
%D A052995 B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 60-61.
%H A052995 Colin Barker, <a href="/A052995/b052995.txt">Table of n, a(n) for n = 0..1000</a>
%H A052995 Younseok Choo, <a href="https://doi.org/10.12988/ijma.2018.81178">Some Results on the Infinite Sums of Reciprocal Generalized Fibonacci Numbers</a>, International Journal of Mathematical Analysis, 12(12) (2018), 621-629.
%H A052995 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H A052995 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A052995 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=1072">Encyclopedia of Combinatorial Structures 1072</a>.
%H A052995 Youngwoo Kwon, <a href="https://arxiv.org/abs/1804.08119">Binomial transforms of the modified k-Fibonacci-like sequence</a>, arXiv:1804.08119 [math.NT], 2018.
%H A052995 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1).
%F A052995 G.f.: -2*x*(-1 + x)/(1 - 3*x + x^2).
%F A052995 a(0) = 0, a(1) = 2, a(2) = 4; for n > 0, a(n) - 3*a(n+1) + a(n+2) = 0.
%F A052995 a(n) = A069403(n-1)+1.
%F A052995 a(n) = Sum(2/5*(-1 + 4*_alpha)*_alpha^(-1-n), _alpha = RootOf(_Z^2 - 3*_Z + 1)).
%F A052995 a(n) = 2*Fibonacci(2*n-1) = 2*A001519(n) for n > 0. - _Vladeta Jovovic_, Mar 19 2003
%F A052995 a(n+2) = F(n)^2 + F(n+3)^2 = 2*F(n+1)^2 + 2*F(n+2)^2, where F = A000045. -  _N. J. A. Sloane_, Feb 20 2005
%F A052995 a(n) = 1/2*(F(2*n+8) mod F(2*n+2)) for n > 2. - _Gary Detlefs_, Nov 22 2010
%F A052995 a(n) = F(n-3)*F(n-1) + F(n)*F(n+2) for n > 0, F(-2) = -1, F(-1) = 1. - _Bruno Berselli_, Nov 03 2015
%F A052995 a(n) = (2^(-n)*((3 - sqrt(5))^n*(1 + sqrt(5)) + (-1 + sqrt(5))*(3 + sqrt(5))^n))/sqrt(5) for n > 0. - _Colin Barker_, Mar 30 2016
%F A052995 a(n) = Fibonacci(2*n-2) + Lucas(2*n-2) for n > 0. - _Bruno Berselli_, Oct 13 2017
%F A052995 a(n) = Lucas(2*n) - Fibonacci(2*n) for n > 0. - _Diego Rattaggi_, Mar 08 2023
%p A052995 spec := [S, S=Prod(Sequence(Union(Prod(Sequence(Z),Z),Z)),Union(Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
%t A052995 LinearRecurrence[{3, -1}, {0, 2, 4}, 30] (* or *)
%t A052995 Nest[Append[#, 3 #[[-1]] - #[[-2]]] &, {0, 2, 4}, 27] (* or *)
%t A052995 CoefficientList[Series[-2 x (-1 + x)/(1 - 3 x + x^2), {x, 0, 29}], x] (* _Michael De Vlieger_, Jul 18 2018 *)
%o A052995 (PARI) concat(0, Vec(2*x*(1-x)/(1-3*x+x^2) + O(x^50))) \\ _Colin Barker_, Mar 30 2016
%o A052995 (PARI) a(n) = fibonacci(max(0,2*n-1))<<1; \\ _Kevin Ryde_, Mar 25 2021
%Y A052995 Bisection of A006355.
%Y A052995 First differences of A025169.
%Y A052995 Cf. A000032, A000045, A055819, A006355, A025169.
%K A052995 nonn,easy
%O A052995 0,2
%A A052995 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052995 More terms from _James Sellers_, Jun 05 2000