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A052995 Expansion of 2*x*(1-x)/(1-3*x+x^2). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 - 18xy + y^2 + 256 = 0. - Colin Barker, Feb 14 2014

REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 30.

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 (2018) Vol. 12, No. 12, 621-629.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

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).

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

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 *)

PROG

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

CROSSREFS

Bisection of A006355.

First differences of A025169.

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

Sequence in context: A095337 A162533 A055819 * A113337 A084575 A081881

Adjacent sequences:  A052992 A052993 A052994 * A052996 A052997 A052998

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

More terms from James A. Sellers, Jun 05 2000

STATUS

approved

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Last modified April 23 18:15 EDT 2019. Contains 322387 sequences. (Running on oeis4.)