A200752 Expansion of (-x^2 + 3*x - 1)/(x^3 - x^2 + 3*x - 1).

1, 0, 0, 1, 3, 8, 22, 61, 169, 468, 1296, 3589, 9939, 27524, 76222, 211081, 584545, 1618776, 4482864, 12414361, 34378995, 95205488, 263651830, 730128997, 2021940649, 5599344780, 15506222688, 42941263933, 118916913891, 329315700428, 911971451326, 2525515567441

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.

a(n+3) is the number of ternary strings of length n in which the number of substrings of the form 0011 equals the number of substrings of the form 11. - John M. Campbell, Nov 02 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2263
• 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 (3,-1,1)

Crossrefs

Cf. A200676, A200739, A200715.

Programs

Maple

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

Formula

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

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

a(n) = 3*a(n-1) -a(n-2) +a(n-3) with a(0)=1, a(1)=0, a(2)=0. - Taras Goy, Jul 23 2017