A052941 Expansion of (1-x)/(1 - 4*x + x^2 + x^3).

1, 3, 11, 40, 146, 533, 1946, 7105, 25941, 94713, 345806, 1262570, 4609761, 16830668, 61450341, 224360935, 819162731, 2990839648, 10919834926, 39869337325, 145566674726, 531477526653, 1940474094561, 7084852176865

Offset: 0

Author

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

Keywords

• easy
• nonn

Comments

a(n) = term (3,1) in M^n, M = the 3 X 3 matrix [1,1,2; 1,2,1; 1,1,1]. - Gary W. Adamson, Mar 12 2009

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 932
• Index entries for linear recurrences with constant coefficients, signature (4,-1,-1).

Programs

GAP

a:=[1,3,11];; for n in [4..30] do a[n]:=4*a[n-1]-a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 18 2019

Magma

I:=[1, 3, 11]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012

Maple

spec:= [S,{S=Sequence(Union(Z,Z,Prod(Union(Sequence(Z),Z),Z)))},unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((1-x)/(1-4*x+x^2+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 18 2019

Mathematica

LinearRecurrence[{4,-1,-1},{1,3,11},30] (* Vincenzo Librandi, Jun 22 2012 *)

PARI

my(x='x+O('x^30)); Vec((1-x)/(1-4*x+x^2+x^3)) \\ Altug Alkan, Sep 21 2018

Sage

def A052941_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-x)/(1-4*x+x^2+x^3)).list()
A052941_list(30) # G. C. Greubel, Oct 18 2019

Formula

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

a(n) = 4*a(n-1) - a(n-2) - a(n-3).

a(n) = Sum_{alpha=RootOf(1-4*z+z^2+z^3)} (3-alpha^2)*alpha^(-1-n)/13.

a(n) = (b(n+2) - b(n+1) + b(n))/13, where b(n) = A356463(n). - Ding Hao, Aug 08 2022

Extensions

More terms from James Sellers, Jun 06 2000