A200676 -id:A200676 - cp's OEIS frontend

A200739 Expansion of (-x^2+5x-1)/(x^3-x^2+5x-1).

1, 0, 0, 1, 5, 24, 116, 561, 2713, 13120, 63448, 306833, 1483837, 7175800, 34701996, 167818017, 811563889, 3924703424, 18979771248, 91785716705, 443873515701, 2146561633048, 10380720366244, 50200913713873, 242770409836169, 1174031855833216, 5677589783043784

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.

Alois P. Heinz, Table of n, a(n) for n = 0..450
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,-1,1)

Cf. A200676.

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

CoefficientList[Series[(-x^2 + 5 x - 1)/(x^3 - x^2 + 5 x - 1), {x, 0, 30}], x] (* or *) LinearRecurrence[{5,-1,1},{1,0,0},30] (* Harvey P. Dale, Nov 26 2017 *)

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

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

A361358 Expansion of x(2 - x)/(1 - 5x + 3*x^2 - x^3).

Original entry on oeis.org

2, 9, 39, 170, 742, 3239, 14139, 61720, 269422, 1176089, 5133899, 22410650, 97827642, 427040159, 1864128519, 8137349760, 35521403402, 155059096249, 676868620799, 2954687218650, 12897889327102, 56302253600359, 245772287239139, 1072852564721720

Offset: 1

Andrew Howroyd, Mar 09 2023

This sequence arises in the enumeration of noncrossing caterpillar graphs (A361356). Given a directed edge (A,B) on the spine of the caterpillar where B is not a leaf node, then a(n) is the number of ways to complete the caterpillar using at most n nodes. Nodes cannot be added to A. Equivalently, a(n) is the number of ways to complete the caterpillar using exactly n nodes allowing leaves to be added to the left of A (but not to the right).

			In the following examples, o is a leaf and 1..n+1 is the spine.
a(1) = 2, a leaf can be added to the left or to the right of the spine:
   1---2    1  o
       |     \ |
       o       2
.
a(2) = a(1) + 7:
   1---2   1---2    1---2    1   o    1   3    1   o   1   o
     /         |      / |      \ |    | / |    |   |   | /
   3---o   o---3    o   o    o---2    2   o    2---3   2---o

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-3,1).

Cf. A361356, A361359, A361360.

LinearRecurrence[{5, -3, 1}, {2, 9, 39}, 30] (* Paolo Xausa, Jul 20 2024 *)

Vec(x*(2 - x)/(1 - 5*x + 3*x^2 - x^3) + O(x^25))

a(n) = 5*a(n-1) - 3*a(n-2) + a(n-3) for n > 3.
a(n) = 2*A200676(n+2) - A200676(n+1).
G.f. A(x) satisfies A(x) = x*(2 - x + 2*A(x))/(1 - x)^3.

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

cp's OEIS Frontend

A200739 Expansion of (-x^2+5*x-1)/(x^3-x^2+5*x-1).

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A361358 Expansion of x*(2 - x)/(1 - 5*x + 3*x^2 - x^3).

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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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A200739 Expansion of (-x^2+5x-1)/(x^3-x^2+5x-1).

A361358 Expansion of x(2 - x)/(1 - 5x + 3*x^2 - x^3).

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).