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

0, 2, 4, 10, 26, 68, 178, 466, 1220, 3194, 8362, 21892, 57314, 150050, 392836, 1028458, 2692538, 7049156, 18454930, 48315634, 126491972, 331160282, 866988874, 2269806340, 5942430146, 15557484098, 40730022148, 106632582346, 279167724890, 730870592324

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• nonn
• easy

Comments

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

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

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

References

• A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.
• B. Muslu, Sayılar ve Bağlantılar 2, Luna, 2021, pages 60-61.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Younseok Choo, Some Results on the Infinite Sums of Reciprocal Generalized Fibonacci Numbers, International Journal of Mathematical Analysis, 12(12) (2018), 621-629.
• Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1072.
• Youngwoo Kwon, Binomial transforms of the modified k-Fibonacci-like sequence, arXiv:1804.08119 [math.NT], 2018.
• Index entries for linear recurrences with constant coefficients, signature (3,-1).

Crossrefs

Bisection of A006355.

First differences of A025169.

Cf. A000032, A000045, A055819, A006355, A025169.

Programs

Maple

spec := [S, S=Prod(Sequence(Union(Prod(Sequence(Z),Z),Z)),Union(Z,Z))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);

Mathematica

LinearRecurrence[{3, -1}, {0, 2, 4}, 30] (* or *)
Nest[Append[#, 3 #[[-1]] - #[[-2]]] &, {0, 2, 4}, 27] (* or *)
CoefficientList[Series[-2 x (-1 + x)/(1 - 3 x + x^2), {x, 0, 29}], x] (* Michael De Vlieger, Jul 18 2018 *)

PARI

concat(0, Vec(2*x*(1-x)/(1-3*x+x^2) + O(x^50))) \\ Colin Barker, Mar 30 2016

PARI

a(n) = fibonacci(max(0,2*n-1))<<1; \\ Kevin Ryde, Mar 25 2021

Formula

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

a(0) = 0, a(1) = 2, a(2) = 4; for n > 0, a(n) - 3*a(n+1) + a(n+2) = 0.

a(n) = A069403(n-1)+1.

a(n) = Sum(2/5*(-1 + 4*_alpha)*_alpha^(-1-n), _alpha = RootOf(_Z^2 - 3*_Z + 1)).

a(n) = 2*Fibonacci(2*n-1) = 2*A001519(n) for n > 0. - Vladeta Jovovic, Mar 19 2003

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

a(n) = 1/2*(F(2*n+8) mod F(2*n+2)) for n > 2. - Gary Detlefs, Nov 22 2010

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

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

a(n) = Fibonacci(2*n-2) + Lucas(2*n-2) for n > 0. - Bruno Berselli, Oct 13 2017

a(n) = Lucas(2*n) - Fibonacci(2*n) for n > 0. - Diego Rattaggi, Mar 08 2023

Extensions

More terms from James Sellers, Jun 05 2000