A200676 Expansion of g.f. -(3*x^2-5*x+1)/(x^3-3*x^2+5*x-1).

1, 0, 0, 1, 5, 22, 96, 419, 1829, 7984, 34852, 152137, 664113, 2899006, 12654828, 55241235, 241140697, 1052634608, 4594992184, 20058197793, 87558647021, 382213633910, 1668450426280, 7283169876691, 31792711738525, 138782499488832, 605817532105276

Offset: 0

Author

Alois P. Heinz, Nov 21 2011

Keywords

• nonn
• easy

Comments

Peter A. Lawrence (see links) has posted a challenge to find a 3x3 integer matrix with "smallish" elements whose powers generate a sequence that is not in the OEIS. This sequence is one of the solutions found.

The sequence without the first 3 entries (1, 5, 22, 96,...) are the partial sums of the partial sums of A049086. - R. J. Mathar, Jul 20 2025

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• D. Birmajer, J. B. Gil, M. D. Weiner, On the Enumeration of Restricted Words over a Finite Alphabet, J. Int. Seq. 19 (2016) # 16.1.3 , example 14.
• Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 7.
• Milan Janjić, Pascal Matrices and Restricted Words, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
• 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 (5,-3,1).

Crossrefs

Cf. A049086, A200739.

Programs

Maple

a:= n-> (<<0|1|0>, <0|0|1>, <1|-3|5>>^n)[1, 1]:
seq(a(n), n=0..30);

Mathematica

CoefficientList[Series[-(3 x^2 - 5 x + 1)/(x^3 - 3 x^2 + 5 x - 1), {x, 0, 26}], x] (* Michael De Vlieger, Sep 04 2018 *)
LinearRecurrence[{5,-3,1},{1,0,0},40] (* Harvey P. Dale, Aug 18 2021 *)

Formula

G.f.: -(3*x^2-5*x+1)/(x^3-3*x^2+5*x-1).

Term (1,1) in the 3x3 matrix [0,1,0; 0,0,1; 1,-3,5]^n.