A353964 -id:A353964 - cp's OEIS frontend

A384646 Expansion of (1+x) / (1-x-5x^2-2x^3).

1, 2, 7, 19, 58, 167, 495, 1446, 4255, 12475, 36642, 107527, 315687, 926606, 2720095, 7984499, 23438186, 68800871, 201960799, 592841526, 1740247263, 5108376491, 14995295858, 44017672839, 129210905111, 379289861022, 1113379732255, 3268250847587, 9593729230906

Offset: 0

Sean A. Irvine, Jun 05 2025

Number of walks of length n starting at vertex 0 in the following graph:
    1---2
   /|\  |
  0 | \ |
   \|  \|
    4---3.
Also, by symmetry, the number of walks of length n starting at vertex 2 in the same graph.

			a(2)=7 because we have the walks 0-1-0, 0-1-2, 0-1-3, 0-1-4, 0-4-0, 0-4-1, 0-4-3.

Sean A. Irvine, Walks on Graphs.
Index entries for linear recurrences with constant coefficients, signature (1,5,2).

Cf. A384647 (vertex 1), A384648 (vertices 3 and 4), A077937 (missing edge {1,3}).

a:= n->  (<<0|1|0|0|1>, <1|0|1|1|1>, <0|1|0|1|0>, <0|1|1|0|1>, <1|1|0|1|0>>^n. <<1,1,1,1,1>>)[1,1]:
seq(a(n), n=0..32);

CoefficientList[Series[(1+x) / (1-x-5*x^2-2*x^3), {x, 0, 32}], x]

a(n) = A353964(n)+A353964(n-1). - R. J. Mathar, Jun 07 2025

A353963 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and right trominoes, n >= 0, k=0..n.

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 1, 1, 13, 47, 1, 1, 45, 259, 3376, 1, 1, 122, 1189, 29683, 475962, 1, 1, 373, 5877, 311894, 9250945, 355724934, 1, 1, 1073, 28167, 3015423, 164776003, 12126673297, 777719132265, 1, 1, 3182, 136723, 30295051, 3051272172, 436744432876, 53090133270415, 6953251175836902

Offset: 0

Gerhard Kirchner, May 13 2022

For tiling algorithm, see A351322.

Reading the sequence {T(n,k)} for k>n, use T(k,n) instead of T(n,k).

			Triangle begins
  n\k  0 1   2    3      4       5         6
  ------------------------------------------
  0:   1
  1:   1 1
  2:   1 1   6
  3:   1 1  13   47
  4:   1 1  45  259   3376
  5:   1 1 122 1189  29683  475962
  6:   1 1 373 5877 311894 9250945 355724934

Liang Kai, Rows n=0..17 of triangle, flattened

Row/columns 0..3 are A000012, A000012, A353964, A353965.
Main diagonal is A354067.
Cf. A351322, A352589.

Maxima
```
See A352589.
```

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

cp's OEIS Frontend

A384646 Expansion of (1+x) / (1-x-5x^2-2x^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A353963 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and right trominoes, n >= 0, k=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

Formula

cp's OEIS Frontend

A384646 Expansion of (1+x) / (1-x-5*x^2-2*x^3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A353963 Triangle read by rows: T(n,k) = number of tilings of a n X k rectangle using 2 X 2 and 1 X 1 tiles and right trominoes, n >= 0, k=0..n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maxima

Formula

A384646 Expansion of (1+x) / (1-x-5x^2-2x^3).