A107785 Sequence obtained using characteristic polynomial that is Laplace transform of the tribonacci characteristic polynomial: -s^4*L(t^3 -t^2 -t -1) = s^3 +s^2 +2*s -6.

0, 1, 1, 3, 7, 5, 37, 69, 35, 395, 739, 261, 4109, 8021, 1763, 42459, 87059, 8437, 437309, 942789, 17549, 4491883, 10183715, 1305243, 46013485, 109726261, 25530749, 470002683, 1179421747, 392600875, 4786258717, 12647990949

Offset

0,4

Links

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

Formula

a(n) = abs(b(n)), where b(n) satisfies b(n) = -b(n-1) - 2*b(n-2) + 6*b(n-3), b(0) = 0, b(1)=b(2)=1.

a(n) = absolute value of the coefficients of the gf x*(1+2*x)/(1 +x +2*x^2 -6*x^3). - G. C. Greubel, May 15 2019

Mathematica

(*first method*) F[1]=0; F[2]=1; F[3]=1; F[n_]:= F[n] = -F[n-1] -2*F[n-2] + 6*F[n-3]; Table[Abs[F[n]], {n, 1, 40}]
(*second method*) M = {{0, 1, 0}, {0, 0, 1}, {6, -2, -1}} v[1] = {0, 1, 1} v[n_] := v[n] = M.v[n - 1] a = Table[Abs[v[n][[1]]], {n, 1, 40}] Det[M - x*IdentityMatrix[3]]
Abs[CoefficientList[Series[x*(1+2*x)/(1+x+2*x^2-6*x^3), {x, 0, 40}], x]] (* G. C. Greubel, May 15 2019 *)

Prog

(PARI) my(x='x+O('x^40)); v=concat([0], Vec(x*(1+2*x)/(1 +x +2*x^2 - 6*x^3))); vector(#v, n, abs(v[n])) \\ G. C. Greubel, May 15 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); b:=Coefficients(R!( x*(1+2*x)/(1+x+2*x^2-6*x^3) ));  [0] cat [Abs(b[n]): n in [1..m-1]];  // G. C. Greubel, May 15 2019
(SageMath) m = 40; T = taylor(x*(1+2*x)/(1+x+2*x^2-6*x^3), x, 0, m); [T.coefficient(x, n).abs() for n in (0..m)] # G. C. Greubel, May 15 2019

Crossrefs

Cf. A000045, A000213.

Sequence in context: A106853 A352011 A083778 * A277207 A380170 A001663

Adjacent sequences: A107782 A107783 A107784 * A107786 A107787 A107788

Keyword

nonn

Author

Roger L. Bagula, Jun 11 2005

Extensions

Edited by G. C. Greubel, May 15 2019

Status

approved