A356463 -id:A356463 - cp's OEIS frontend

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

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

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

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).

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

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

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

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

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

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

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

More terms from James Sellers, Jun 06 2000

A356411 Sum of powers of roots of x^3 - x^2 - x - 3.

Original entry on oeis.org

3, 1, 3, 13, 19, 41, 99, 197, 419, 913, 1923, 4093, 8755, 18617, 39651, 84533, 180035, 383521, 817155, 1740781, 3708499, 7900745, 16831587, 35857829, 76391651, 162744241, 346709379, 738628573, 1573570675, 3352327385, 7141783779

Offset: 0

Greg Dresden, Aug 05 2022

a(n) is the sum of the n-th powers of the three roots of x^3 - x^2 - x - 3. These roots are c1 = 2.130395..., c2 = -0.5651977... - i*1.0434274..., and c3 = -0.5651977... + i*1.0434274..., and so a(n) = c1^n + c2^n + c3^n. The real parts of c2 and c3 are A273065.

a(n) can also be determined by Vieta's formulas and Newton's identities. For example, a(3) by definition is c1^3 + c2^3 + c3^3, and from Newton's identities this equals e1^3 - 3*e1*e2 + 3*e3 for e1, e2, e3 the elementary symmetric polynomials of x^3 - x^2 - x - 3. From Vieta's formulas we have e1 = 1, e2 = -1, and e3 = 3, giving us e1^3 - 3*e1*e2 + 3*e3 = 1 + 3 + 9 = 13, as expected.

			For n=3, a(3) = (2.130395...)^3 + (-0.5651977... - i*1.0434274...)^3 + (-0.5651977... + i*1.0434274...)^3 = 13.

Index entries for linear recurrences with constant coefficients, signature (1,1,3).

Cf. A103143, A123102, A247594, A356463, A273065 (Re c2,c3).

LinearRecurrence[{1, 1, 3}, {3, 1, 3}, 40]

polsym(x^3 - x^2 - x - 3, 35) \\ Joerg Arndt, Aug 11 2022

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

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

cp's OEIS Frontend

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

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