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A052995 Expansion of 2x(1-x)/(1-3x+x^2). 7
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 (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

Table of n, a(n) for n=0..27.

Guo-Niu Han, Enumeration of Standard Puzzles

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1072

Index to sequences with linear recurrences with constant coefficients, signature (3,-1).

FORMULA

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

Recurrence: {a(0)=0, a(2)=4, a(1)=2, a(n)-3*a(n+1)+a(n+2)=0}.

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), n>0. - Vladeta Jovovic, Mar 19 2003

a(n+2) = F(n)^2 + F(n+3)^2 = 2F(n+1)^2 + 2F(n+2)^2.

a(n) = 1/2*(fib(2n+8) mod fib(2n+2)), n>2.

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);

CROSSREFS

Equals A069403(n-1)+1. Bisection of A006355. First differences of A025169. Cf. A055819.

Sequence in context: A095337 A162533 * A055819 A113337 A084575 A081881

Adjacent sequences:  A052992 A052993 A052994 * A052996 A052997 A052998

KEYWORD

easy,nonn

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 July 30 13:12 EDT 2014. Contains 245069 sequences.