A052550 Expansion of (1-2*x)/(1 - 3*x - x^2 + 2*x^3).

1, 1, 4, 11, 35, 108, 337, 1049, 3268, 10179, 31707, 98764, 307641, 958273, 2984932, 9297787, 28961747, 90213164, 281005665, 875306665, 2726499332, 8492793331, 26454265995, 82402592652, 256676457289, 799523432529, 2490441569572

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 487
• Index entries for linear recurrences with constant coefficients, signature (3,1,-2).

Programs

GAP

a:=[1,1,4];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, May 07 2019

Magma

I:=[1,1,4]; [n le 3 select I[n] else 3*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..30]]; // G. C. Greubel, May 07 2019

Maple

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

Mathematica

LinearRecurrence[{3,1,-2}, {1,1,4}, 30] (* G. C. Greubel, May 07 2019 *)

PARI

my(x='x+O('x^30)); Vec((1-2*x)/(1-3*x-x^2+2*x^3)) \\ G. C. Greubel, May 07 2019

Sage

((1-2*x)/(1-3*x-x^2+2*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 07 2019

Formula

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

a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3), with a(0)=1, a(1)=1, a(2)=4.

a(n) = Sum((-1/229) * (-5 - 74*alpha + 16*alpha^2) * alpha^(-1-n), alpha = RootOf(1 - 3*z - z^2 + 2*z^3)).

Extensions

More terms from James Sellers, Jun 06 2000