A052936 Expansion of (1-x)*(1-2*x)/(1-5*x+5*x^2).

1, 2, 7, 25, 90, 325, 1175, 4250, 15375, 55625, 201250, 728125, 2634375, 9531250, 34484375, 124765625, 451406250, 1633203125, 5908984375, 21378906250, 77349609375, 279853515625, 1012519531250, 3663330078125, 13254052734375

Offset

0,2

Comments

For n>=0, a(n) is the number of generalized compositions of n+1 when there are 2^(i-1)+2 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 925

Index entries for linear recurrences with constant coefficients, signature (5,-5).

Formula

G.f.: (1-x)*(1-2*x)/(1-5*x+5*x^2).

a(0)=1, a(1)=2, a(2)=7, a(n) = 5*a(n-1) - 5*a(n-2). - Harvey P. Dale, Apr 20 2012

a(n) = Sum_{alpha=RootOf(1-5*z+5*z^2)} (1/5)*(1 - alpha)*alpha^(-1-n).

The sequence beginning 2, 7, 25 ... has g.f. (2-3*x)/(1-5*x+5*x^2), a(n) = (1-2/sqrt(5))*(5/2-sqrt(5)/2)^n + (5/2+sqrt(5)/2)^n*(1+2/sqrt(5)). It is the binomial transform of Fib(2*n+3) and the second binomial transform of Fib(n+3). Also, its n-th term is the n-th term of the 3rd binomial transform of Fib(3*n+3) divided by 2^n. - Paul Barry, Mar 23 2004

Binomial transform of convolution of Fib(2*n+1) and (-1)^n. Binomial transform of Fib(n+1)^2. - Paul Barry, Sep 27 2004

a(n) = Sum_{k=0..n} C(n-1, k)*Fib(2*n-2*k+1). - Paul Barry, Jun 07 2005

a(2*n) = 5^(n-1)*Lucas(2*n+2) for n>=1 with a(2*0)=1 and a(2*n+1) = 5^n * Fibonacci(2*n+3). - G. C. Greubel, Oct 18 2019

Maple

spec:= [S, {S=Sequence(Prod(Union(Sequence(Z), Sequence(Union(Z, Z))), Z))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);
seq(coeff(series((1-x)*(1-2*x)/(1-5*x+5*x^2), x, n+1), x, n), n = 0 .. 30); # G. C. Greubel, Oct 18 2019

Mathematica

Join[{1}, LinearRecurrence[{5, -5}, {2, 7}, 30]] (* Harvey P. Dale, Apr 20 2012 *)
Join[{1}, Table[If[EvenQ[n], 5^(n/2 -1)*LucasL[n+2], 5^((n-1)/2)* Fibonacci[n + 2]], {n, 30}]] (* G. C. Greubel, Oct 18 2019 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((1-x)*(1-2*x)/(1-5*x+5*x^2)) \\ G. C. Greubel, Oct 18 2019
(Magma) I:=[2, 7]; [1] cat [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // G. C. Greubel, Oct 18 2019
(Sage)
def A052936_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P((1-x)*(1-2*x)/(1-5*x+5*x^2)).list()
A052936_list(30) # G. C. Greubel, Oct 18 2019
(GAP) a:=[2, 7];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; Concatenation([1], a); # G. C. Greubel, Oct 18 2019

Crossrefs

Cf. A000032, A000045.

Sequence in context: A070859 A048576 A018907 * A108152 A024482 A097613

Adjacent sequences: A052933 A052934 A052935 * A052937 A052938 A052939

Keyword

easy,nonn

Author

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

Extensions

More terms from James A. Sellers, Jun 06 2000

Status

approved