A200739 -id:A200739 - cp's OEIS frontend

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

Alois P. Heinz, Nov 21 2011

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

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).

Cf. A049086, A200739.

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

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 *)

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.

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

Original entry on oeis.org

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

Alois P. Heinz, Nov 21 2011

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

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)

Cf. A200676, A200739, A200715.

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

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

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

Original entry on oeis.org

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

Alois P. Heinz, Nov 21 2011

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, ...}.

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)

Cf. A200676, A200739.

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

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 *)

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

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

A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

Original entry on oeis.org

1, 1, 13, 47, 259, 1189, 5877, 28167, 136723, 660173, 3194613, 15445007, 74699811, 361230229, 1746933205, 8448061879, 40854753875, 197572345789, 955455626773, 4620559362303, 22344915889827, 108059470995013, 522573007884725, 2527150465444071, 12221238828079379

Offset: 0

Gerhard Kirchner, May 13 2022

For tiling algorithm see A351322.

			a(2) = 13:
    v    h,v   h=v   h,v
   ___   ___   ___   ___   ___
  |   | | |_| |  _| |  _| |_|_|    mirroring included
  |___| |___| |_| | |_|_| |_|_|    h: horizontal, v: vertical
  |_|_| |_|_| |___| |_|_| |_|_|
    2  +  4  +  2  +  4  +  1 = 13

Index entries for linear recurrences with constant coefficients, signature (3,9,-1,2).

Cf. A351322, A352589, A353963, A353964.

Maxima
```
See A352589.
```

G.f.: (1 - 2*x + x^2) / (1 - 3*x - 9*x^2 + x^3 - 2*x^4).
a(n) = 3*a(n-1) + 9*a(n-2) - a(n-3) + 2*a(n-4).
31*a(n) = 18*(-2)^n +13*A200739(n+3) +2*A200739(n+2) +9*A200739(n+1). - R. J. Mathar, Jun 07 2025

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

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

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

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A353965 Number of tilings of a 3 X n rectangle using 2 X 2 and 1 X 1 tiles and right trominoes.

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

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

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