A052965 Expansion of (1-x)/(1-3x-4x^2+4x^3).

1, 2, 10, 34, 134, 498, 1894, 7138, 26998, 101970, 385350, 1455938, 5501334, 20786354, 78540646, 296762018, 1121303222, 4236795154, 16008550278, 60487618562, 228549876182, 863565901682, 3262946735526, 12328904308578

Offset

0,2

Links

Table of n, a(n) for n=0..23.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1036

Index entries for linear recurrences with constant coefficients, signature (3,4,-4).

Formula

G.f.: -(-1+x)/(1-3*x-4*x^2+4*x^3).

Recurrence: {a(0)=1, a(1)=2, a(2)=10, 4*a(n)-4*a(n+1)-3*a(n+2)+a(n+3)=0}.

Sum(-1/158*(-17-49*_alpha+40*_alpha^2)*_alpha^(-1-n), _alpha=RootOf(1-3*_Z-4*_Z^2+4*_Z^3)).

Limit_{n->oo} a(n)/a(n-1) = 2+A316139. - Bruce Nye, Jan 23 2026

Maple

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

Mathematica

CoefficientList[Series[(1-x)/(1-3x-4x^2+4x^3), {x, 0, 40}], x] (* or *) LinearRecurrence[{3, 4, -4}, {1, 2, 10}, 40] (* Harvey P. Dale, Dec 01 2016 *)

Prog

(PARI) Vec((1-x)/(1-3*x-4*x^2+4*x^3)+O(x^25)) \\ Bruce Nye, Jan 23 2026

Crossrefs

Cf. A316139.

Sequence in context: A124634 A306099 A192378 * A108924 A281097 A221492

Adjacent sequences: A052962 A052963 A052964 * A052966 A052967 A052968

Keyword

easy,nonn

Author

INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000

Extensions

More terms from James Sellers, Jun 06 2000

Status

approved