A200715 Expansion of (-3*x^2 + x - 1)/(x^3 - 3*x^2 + x - 1).

1, 0, 0, 1, 1, -2, -4, 3, 13, 0, -36, -23, 85, 118, -160, -429, 169, 1296, 360, -3359, -3143, 7294, 13364, -11661, -44459, 3888, 125604, 69481, -303443, -386282, 593528, 1448931, -717935, -4471200, -868464, 11827201, 9961393, -26388674, -44445652, 44681763

Offset: 0

Author

Alois P. Heinz, Nov 21 2011

Keywords

• sign
• easy

Comments

Peter A. Lawrence (see links) has posted a challenge to find a 3 X 3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.

|a(n)| is a prime number for n in {5, 7, 8, 11, 19, 27, 108, 276, 371, 608, ...} with values {2, 3, 13, 23, 3359, 69481, 167527749243856707416101, ...}.

Links

• Alois P. Heinz, 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.
• Peter Lawrence et al., sequence challenge and follow-up messages on the SeqFan list, Nov 21 2011
• Index entries for linear recurrences with constant coefficients, signature (1,-3,1)

Crossrefs

Cf. A200676, A200739.

Programs

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|1>>^n)[1, 1]:
seq(a(n), n=0..50);

Mathematica

CoefficientList[Series[(-3x^2+x-1)/(x^3-3x^2+x-1),{x,0,40}],x] (* or *) LinearRecurrence[{1,-3,1},{1,0,0},40] (* Harvey P. Dale, Nov 22 2011 *)

PARI

Vec((-3*x^2+x-1)/(x^3-3*x^2+x-1)+O(x^99)) \\ Charles R Greathouse IV, Nov 22 2011

Formula

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

Term (1,1) in the 3 X 3 matrix [0,1,0; 0,0,1; 1,-3,1]^n.

a(n) = a(n-1) - 3*a(n-2) + a(n-3); a(0)=1, a(1)=a(2)=0. - Harvey P. Dale, Nov 22 2011